#5751 Praxis - Mathematics Terms 2020
Find the Slope of the line that goes through ( 4, 2) and ( -3, 16).
( 4, 2) and (-3, 16) M (slope) = y2 -y1 / x2 - x1 So Start is (4,2) End: (-3, 16) -3 - 4 = -7 Run 16 - 2 = 14 Rise Simplify to get -𝟐 𝐬𝐥𝐨𝐩𝐞.
Prove these two triangles congruence: Where side DC∥AB and they share a side AC.
Prove: ΔDCA ≅ΔBAC There is a transversal line going between the two triangles so: ∠CAB = ∠ACD Given the information that we have, we CANNOT prove congruency.
Long Division
Use to either get a whole number and a remainder and go from left to right finding the greatest multiple of the dividing number. Add a O and a decimal to get the value without remainder or leave remainder.
Circumcenter of a Triangle
The point of concurrency of all sides of the triangle on the perpendicular bisector .
Geometric Dilation
When I geometric Shape is scaled up or scaled down.
Circumcircle
A circle formed around a triangle with a circumcenter where all vertices lay on the circumference of the circle.
Ratio (quotient)
A comparison of two quantities ex: 5 fruits, 2 apples: 3 oranges 2:3 or 2/5 and 3/5
Parallelogram
A convex quadrilateral that MUST HAVE two pairs of parallel lines and congruent angles.
Congruent Angles
Angles that have the same measurement.
Draw the Image of ΔABC under a dilation whose center is P and scale factor is 1/4: where: CB ~ 5.7 BA ~ 14.4 AC ~ 14.4
CB ~ 5.7 BA ~ 14.4 AC ~ 14.4 Scale factor is 1/4 P is the center of the Dilation. Therefore: The distance from C and B to P is 8 and the distance from A to P is 4, so 1/4 of all three means that: A(1,-1) B(-1, 2) C(-2,1)
Bisector
The ray that divides the angle into two congruent(identical in length) adjacent (side-by-side) angles. Ex: ∠AMC =∠BMC These are equal measurements made by making a congruent bisector (even line through the middle).
A Sphere's Center
The set of ALL points in a 3D space that are equidistant to the center point.
Isosceles Triangle
an isosceles triangle is a triangle that has two sides of equal length.
Solve this 2-Step Inequality: 2/3 > -4Y - 8 and 1/3
2/3 > -4Y - 8 and 1/3 Isolate the Y and don't flip the sign. 2/3 + 25/3 > -4Y 27/3 = 9 > -4Y Isolate the Y, now you must FLIP. 9 (-4) ....Y --- < ------- -4 ........-1/4 (they cancel out) So: (-𝟗/𝟒) < 𝐘 Simplify: -𝟐 𝐚𝐧𝐝 𝟏/𝟒 < 𝐘
Solve Linear Equation: 2X + 3 = -15
2x + 3 = -15 Get the variable alone by subtracting 3 from both sides: 2x = -18 Get X alone so divide by 2: 𝐗 = -𝟗 That is how you get a straight horizontal line if you were graphing. Check it: 2(-9) + 3 = -15 -𝟏𝟖 + 𝟑 = -𝟏𝟓 𝐓𝐡𝐢𝐬 𝐢𝐬 𝐜𝐨𝐫𝐫𝐞𝐜𝐭!
Rhombus
A convex quadrilateral that has two pairs of parallel lines, two congruent angles, AND all sides have equal lengths.
Convex Quadrilateral
A four sided shape with four sides, four angles, and four vertex who's angles do NOT exceed 180°.
Finding Triangle angles using intersecting lines: Given angles 121 on a line, 29 on another, and 29 on another, what is the value of X: Refer to the image:
By using the data given we know that all straight lines and triangles have supplementary angles. So for green: 180 = 121 - A A = 59 For the blue: 29 - B = 180 B = 151 For the Red you need to complete the triangle angles created by blue and green supplementary angles: 59 + 29 + Z = 180 Z = 92 Then you find the missing supplementary angle to the lower blue angled triangle: 92 + 29 = 59 Now you have to subtract the adjacent angle to find the missing value for X so: 59 - Y = 180 Adjacent angle (Y) = 121 Therefore: 121 - X = 180 and X = 59°
Two-Way Relative Frequency Tables
Of 125 SUVs, 28 had accidents, 97 had no accidents. Of 139 SP Cars, 35 had accidents, ad 104 had no accidents. So 28/125 Accidents were SUVS and 35/104 were SP Cars so divide the ratios of accidents nbny the type of vehicle and we have 22% of all drivers of SUV got into accidnets and 25% all Sp Cars got into accidents.
How do you get a Positive Root from a Negative Number in the Radical Sign?
√-4 we need to find the positive so, (i) is the value of √-1 since it is an imaginary #. ' √(-1) = √ i √𝟒 * √ 𝐢 = 𝟐𝐢
Factors
𝐈𝐧𝐭𝐞𝐠𝐞𝐫𝐬 𝐭𝐡𝐚𝐭 𝐰𝐞 𝐜𝐚𝐧 𝐦𝐮𝐥𝐭𝐢𝐩𝐥𝐲 𝐭𝐨 𝐠𝐞𝐭 𝐭𝐡𝐞 𝐜𝐡𝐨𝐬𝐞𝐧 #. Find the Factors of 12: 1, 2, 3, 4, 6, 12 All can be multiplied to get the number 12
Radicals
"𝐑𝐨𝐨𝐭" or opposite of exponents. EX: 2√ is Square Root. 3√ is Cube Root 4√ is 4th Root etc.
Proportion
An 𝐞𝐪𝐮𝐚𝐥𝐢𝐭𝐲 𝐛𝐞𝐭𝐰𝐞𝐞𝐧 𝐭𝐰𝐨 𝐫𝐚𝐭𝐢𝐨𝐬 or quantities that can be written as a fraction or as a ratio without changing it's value. ex: 1/4 = 2/8
Integer Exponents
Integers (whole numbers) + / - that indicate the # of self multiplications. ex: X^2 = x*x
How do you find the measure of angles formed by transversal lines if: ∠BEF = 6X + 182 ∠CBE = 9X + 88
Lines ABC ∥ DEF and line GH is transverse (intercepting both other lines). We know that lines make supplemental angles so they MUST equal to 180°. ∠BEF = 6X + 182 ∠CBE = 9X + 88 First combine similar terms: 180° = 6X + 9X + 182 + 88 180 =...15X + 270 -180 = -15X -180 ----------------- -15X = 90 ---------- .......-15 X = -6 Now plug into the formulas to get m∠BEF and m∠CBE : ∠BEF = 6X + 182 ∠BEF = 6(-6) + 182 ∠BEF = -36 + 182 ∠𝐁𝐄𝐅 = 𝟏𝟒𝟔° ∠CBE = 9X + 88 ∠CBE = 9(-6) + 88 ∠CBE = -54 + 88 ∠𝐂𝐁𝐄 = 𝟑𝟒°
How do you use three points to define a circle?
Looking at the triangle we know that: AB = AL + BL BC = BM + MC AC = AK + KC and that They are all equidistant to 0. 0 = Circumcenter equal vertices A, B, C. Cr= Circumradius. AND Cr = OA = OC = OB Therefore this Triangle is the circumcenter of a circle.
Co-linear
Multiple points that sit on the same segment of a line.
Solving Linear Equations to Graph: Y = 2X - 3 All Linear Equations form a line.
Solve this linear Equation: Y = 2X - 3 Create a T - Chart to calculate where the line will be: X Y 0 , -3 Y = 2(0) - 3 = -3 (X-Intercept) 1 , -1 Y = 2(1) - 3 = -1 2 , 1 Y = 2(2) - 3 = 1 𝐒𝐨 𝐬𝐥𝐨𝐩𝐞 (𝐘) 𝐢𝐬 𝐑𝐢𝐬𝐞 𝟐 𝐨𝐯𝐞𝐫 𝐑𝐮𝐧 𝟏 . 𝐘 = 𝟐/𝟏
Multiples
The "Result" of multiplying the chosen # by another integer. (whole number) 𝐄𝐗: 𝐌𝐮𝐥𝐭𝐢𝐩𝐥𝐞𝐬 𝐨𝐟 𝟔 𝐚𝐫𝐞 𝟎,𝟔,𝟏𝟐,𝟏𝟖,𝟐𝟒 𝐞𝐭𝐜..
Circumradius
The Radial length of the circumcenter (Center vertex to all points lying on the circumference of the circle
A Circle's Center
The Vertex of any two line segments where one End-point is C for center, and the others are found along the equidistant perimeter.
Concurrent Angle of a Triangle's Bisector
The angle at which the bisector intersects a line or shape. Aka the middle of two corners of a triangle.
Use Degenerate Triangles make the Triangle Inequality True for Triangle side C: where: A = 10 , B = 6, C = X
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. The length of side C must be greater than the product of the mid length subtracted from the longest side. a = 10 b = 6 c = x Therefore: A + B ≥ C a(10) + b(6 )≥ c(16) AND A - B ≤ C a(10) - b(6) ≤ c(4)
Quantitative Reasoning
The 𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐭𝐨 𝐦𝐞𝐚𝐬𝐮𝐫𝐞 𝐚 𝐯𝐚𝐥𝐮𝐞 using algebra, geometry and data analysis such as statistics, ratios and probability. Opposite of Qualitative reasoning: can't be measured mathematically.
Degenerate Triangle
Three co-linear points (Sharing one line) that is technically not a triangle because all three points are on the same line now. Missing length C - (anywhere between A-B or A+B) C can be as low as A-B=C OR A+B=C Ex: A = 10 B = 6 C = 10+6=X OR 10-6=X Side C must be: 4 < C < 16
Vertical Angle
Vertical angles are the angles opposite each other( reciprocal) when two lines cross (intersect). ∠ = Angle m= Measure Line AB and Line CD that intersect at point E. THEREFORE: Line AEB and it's reciprocal, Line CED = 180° (Straight). AND: m∠BED and it's reciprocal m∠CEA = 70° THEREFORE: m∠CEB and it's reciprocal m∠AED = 110° SO: 𝐦∠𝐁𝐄𝐃 (𝟕𝟎°) + 𝐦∠𝐂𝐄𝐀 (𝟕𝟎°) = 𝐕𝐞𝐫𝐭𝐢𝐜𝐚𝐥 𝐀𝐧𝐠𝐥𝐞𝐬 𝐦∠𝐂𝐄𝐁 (𝟏𝟏𝟎°) + 𝐦∠𝐀𝐄𝐃 (𝟏𝟏𝟎°) = 𝐕𝐞𝐫𝐭𝐢𝐜𝐚𝐥 𝐀𝐧𝐠𝐥𝐞𝐬
Geometric Reflection
When a geometric shape reflects or flips over a certain line. The reflection line will be seen as "l' or lowercase L. Ex: Reflect a(3,6) over line L that is vertical sitting on the (0.0) point means it will be (-3,6) because it is sitting on the 0 Vertically.
Which Ordered Pair is a Solution to: -3X -Y = 6 a - Only (-4, 4) b - Only (-3, 3) c - Both d - Neither
Which Ordered Pair is a Solution to: -3X -Y = 6 a - Only (-4, 4) b - Only (-3, 3) c - Both d - Neither Plug in the X and Y values to find the values that are the solution: a - only (-4, 4) (-3)(-4) - (4) = 6 12 - 4 does not equal 6 so NO b - only ( -3, 3) (-3)(-3) -(3) = 6 9 - 3 does equal 6 so YES 𝐁 𝐢𝐬 𝐭𝐡𝐞 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧!
What does this term mean? XZ =ZY
XZ = ZY Means that the line segments between these two segments on an unending line are equal to one another. Therefore, 𝐙 𝐢𝐬 𝐭𝐡𝐞 𝐦𝐢𝐝𝐩𝐨𝐢𝐧𝐭, or halfway between points X and Y.
End-Points
the points where the line segment ends which is non-dimensional. Labelled with Letters like A, B, etc.
Integer
𝐖𝐡𝐨𝐥𝐞 𝐧𝐮𝐦𝐛𝐞𝐫𝐬 that can be positive ( + ) or ( - ) negative. Ex: -295, -2, 5, 1894.
Quadratic Equation
𝐚𝐱² + 𝐛𝐱 + 𝐜 = 𝟎 *the format in which many algebra questions come in. USE QUADRATIC FORMULA TO SOLVE
Solve the Linear Equation where: (X + 2) / (X + 1) = 7
(X + 2) -------- = 7 (X + 1) First, Get the X out of the Denominator: (X + 2) x (X + 1) --------------- = 7 (X + 1) (X + 1) Then you get: X + 2 = 7(X + 1) ....X + 2 = ..7X + 7 -7X .-2 = -7X - 2 ---------------------- .......-6X = 5 Completely Isolate the X: -6X = 5....................................................... 𝟓 ---- = --- So that 𝐗 = (𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞) - ---- -6 ......-6 .............................................................𝟔
Solve the Linear Equation where: 3/X = 5 *This time the X is in the denominator*
3/X = 5 First get the X out of the Denominator: X times 3/X = 5 times X So 5X = 3 Isolate the x completely: 5 X ... 3 ---..=..--- ...5..... 5 𝐒𝐨 𝐗 = 𝟑/𝟓
Angle
A figure composed of two rays or line segments that meet at a common point, or vertex. Shown as ∠
What is a kite as a geometric shape?
A kite is a convex quadrilateral and has two pairs of congruent sides adjacent from one another. Any rhombus can be a kite.
Ray
A part of a line, with one endpoint (point A for example), that continues without end in one direction, even past another point. Ex: AD (Add an arrow above AD indicating which way the ray is moving).
Line Segment
A part of a 𝐥𝐢𝐧𝐞 𝐭𝐡𝐚𝐭 𝐡𝐚𝐬 𝐭𝐰𝐨 𝐞𝐧𝐝𝐩𝐨𝐢𝐧𝐭𝐬 which is one dimensional and any and all points in a straight line between both end point. Written AB (with a line over AB).
Midpoint
A point that divides a segment into two congruent segments.
Line
A straight one dimensional line that does not end. Ex: EF (Insert line with arrows on either end above EF).
Transversal Line
A transversal line is a 𝐥𝐢𝐧𝐞 𝐭𝐡𝐚𝐭 𝐢𝐧𝐭𝐞𝐫𝐬𝐞𝐜𝐭𝐬 𝐦𝐮𝐥𝐭𝐢𝐩𝐥𝐞 𝐥𝐢𝐧𝐞𝐬.. Ex: (A - B) ∥ (C - D) Both AB and CD are parallel to one another. (E - F) is a line that intersects both AB and CD.
Scientific Notation
A way in which large numbers are condensed using exponents. EX: 4,900,000,000 = 𝟒.𝟗 𝐱 𝟏𝟎^𝟗. * the power always equals however many places you need to move over to get to 4.9 that you would x by 10.
Pentagon A'B'C'D'E' is the image of pentagon ABCDE under a dilation with a scale factor of 5/2. What is the length of segment A'E'?
Because there is no specific dilation amount, we can only figure out the respective lengths. If the scale factor is 5/2 we know that we must multiply what we think the length values are by 2.5. Therefore: AE = 5
Using Transverse Lines, prove that the interior angle values of X, Y, and Z = 180° In this example, Light Blue Line ∥ Orange Line
Because these are transverse lines we know that: P = Purple Line O = Orange Line B = Blue line So if ∠PO = X then ∠PB and it's transverse must also = X And if ∠RO = Y then ∠BR = Y as well. And if ∠PR = Z then it's vertical angle ∠PR must also be the value of X. Looking at the Blue line we know that X + Y + Z = 180°
Pentagon A'B'C'D'E' is the image of pentagon ABCDE under dilation. What is the scale factor of the dilation?
By finding the slope distance between points in each line segment you can find out if all factors are the same. BC = (3,0) B'C' = (1,0) CD = (9,-6) C'D'= (3,-2) DE = (-9,-3) D'E' = (-3,-1) EA = (-3,3) E'A' = (-1,1) AB = (0,6) A'B' = (0,2) What is the GCF of all of thoe X and Y prime and regular points? 3 Line segment BC is (3,0) and (1,0) giving it a factor of 3. Or if CD is (9,-6) then CD' would be (3,-2) which is correct!
Which of the ordered pairs is a solution to the following equation: 4X - 1 = 3Y + 5 a - Only ( 3, 2) b - Only ( 2, 3) c - Both d - Neither
Check all options where there is a solution: 4X - 1 = 3Y + 5 a - Only ( 3, 2) 4(3) -1 = 3(2) + 5 12 - 1 = 6 + 5 11 = 11 so Yes, 𝐀 𝐢𝐬 𝐚 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧! b - Only ( 2, 3) 4(2) -1 = 3(3) + 5 8 -1 = 9 + 5 7 does not equal 14 so NO 𝐁 𝐢𝐬 𝐧𝐨𝐭 𝐚 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧. c - Both d - Neither 𝐓𝐡𝐞 𝐚𝐧𝐬𝐰𝐞𝐫 𝐢𝐬 𝐀, 𝐨𝐧𝐥𝐲 ( 𝟑, 𝟐) 𝐢𝐬 𝐚 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧!
What is Dependent Probability and how does it affect latter outcomes?
Dependent Probability is the when the 𝐨𝐮𝐭𝐜𝐨𝐦𝐞 𝐨𝐟 𝐭𝐡𝐞 𝐩𝐫𝐞𝐯𝐢𝐨𝐮𝐬 𝐢𝐧𝐬𝐭𝐚𝐧𝐜𝐞 𝐨𝐫 𝐞𝐯𝐞𝐧𝐭 𝐚𝐟𝐟𝐞𝐜𝐭𝐬 𝐭𝐡𝐞 probability of getting the desired 𝐨𝐮𝐭𝐜𝐨𝐦𝐞 𝐢𝐧 𝐭𝐡𝐞 𝐧𝐞𝐱𝐭 𝐞𝐯𝐞𝐧𝐭. For example, after drawing a green marble out of a satchel, you don't put it back and try again to get another green marble. Your original probability at this first draw would be 60% Green or 3:5 When You remove the Green marble, you lose a chance at re-picking it because you don't put it back so 𝐢𝐭 𝐥𝐨𝐰𝐞𝐫𝐬 𝐭𝐡𝐞 𝐭𝐨𝐭𝐚𝐥 𝐩𝐨𝐬𝐬𝐢𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬 𝐚𝐬 𝐰𝐞𝐥𝐥 𝐚𝐬 𝐭𝐨𝐭𝐚𝐥 𝐩𝐞𝐫𝐜𝐞𝐧𝐭𝐚𝐠𝐞 𝐚𝐯𝐚𝐢𝐥𝐚𝐛𝐥𝐞 desired outcomes. 2:4 or 50%
Find the Exterior Angles of an Convex (no dents in the shape) polygon with 5 different angles.
Imagine cutting a pie with 5 slices. Each corner of the polygon is an opportunity to extend the angle lines and make a complete 360. A = Green B = Blue C = Purple D = Red E = Orange A + B + C + D + E = 360° Imagine cutting a pie with 5 slices.
How do you express (write) congruence in triangles? Use Δ and ≅
Looking at the image provided: Δ = Triangle ≅ = congruent (identical) Δ AMC ≅ Δ MBC AC = BC Show in S.A.S Congruence. SAS = Side, Angle, Side These are equal measurements made by making a congruent bisector (even line through the middle) of a Side (length) or Angle.
Geometric Dialation:
Stretching or 'scaling up' a point's current position or a shape's size.
Vertex
The starting point or common point between two or more line segments or rays.
Statistic and Probability Word Question: The marching band is holding a raffle at a football game, with two prizes. After the first ticket is pulled out and the winner determined, the ticket is taped to the prize. The next ticket is pulled out to determine the winner of the second prize. Are the two events independent? Explain.
The two events are 𝐝𝐞𝐩𝐞𝐧𝐝𝐞𝐧𝐭 compounding events because they are two events that come from the same populous and the 𝐫𝐞𝐦𝐨𝐯𝐚𝐥 𝐨𝐟 𝐭𝐡𝐞 𝐰𝐢𝐧𝐧𝐞𝐫'𝐬 𝐭𝐢𝐜𝐤𝐞𝐭 𝐜𝐡𝐚𝐧𝐠𝐞𝐬 𝐭𝐡𝐞 𝐩𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲 and betters it ever so slightly simply by making the pool smaller. Let's say that there are 100 ticket holders for the 2 winning prizes. There is a 1% chance that you will win the first prize on the first draw. On the second draw, there are only 99 other ticket holders and therefore there is a 1.01% chance that you will win the second prize. Because the winning tickets were not picked together, the chance would not exceed 1% on the first draw because at that time they were only pulling for 1 prize. It is only AFTER the first winner is announced that the second prize even begins to become a possibility. 𝐅𝐢𝐫𝐬𝐭 𝐏𝐫𝐢𝐳𝐞: 𝟏% 𝐂𝐡𝐚𝐧𝐜𝐞 𝐚𝐭 𝐚 𝐩𝐫𝐢𝐳𝐞. 𝐒𝐞𝐜𝐨𝐧𝐝 𝐏𝐫𝐢𝐳𝐞: 𝟏.𝟎𝟏% 𝐂𝐡𝐚𝐧𝐜𝐞 𝐚𝐭 𝐚 𝐩𝐫𝐢𝐳𝐞.
Acute Angle
Two rays that come from a common point, or Vertex or center (of a circle), who's interior angle is less than 90°. Ex: Imagine a pie who's complete angle is 360°, and there are 6 slices. Each slice from tip to crust (two line segments that share the same vertex) would be 60°. 60° times 6 slices would equal 360°, so each slice would be less than 90°, making the slice an acute angle. This is comparable to when something or someone is sharp, they have an acute sense of what is going on. 𝐀𝐜𝐮𝐭𝐞 < 𝟗𝟎° 𝐨𝐫 ∠𝐀𝐁𝐂 = 𝟑𝟎°
What is the relationship between the area of a triangle and it's circumscribed circle or circumcircle.
We know: h = height b = base d = diameter cc = circumcenter AB = AD +DB BC = BE + EC AC = AF + FC ∠BFC = 90° ∠AFB = 90° The area will be half the base times height, and we can find the area of a triangle using the circumcircle or the arc angle!
What happens if there is a negative coefficient or you have to divide to isolate a variable in a linear inequality?
You must 𝐅𝐋𝐈𝐏 𝐭𝐡𝐞 𝐢𝐧𝐞𝐪𝐮𝐚𝐥𝐢𝐭𝐲 because 1 < 2 but -1 > -2. Changing the sign changes the value and -2 is smaller than -1 on a line chart, therefore the sign must be switched.
What is the formula to find the Area of a triangle? Area of a Triangle = aΔ Height = h Base = b
aΔ = (.5)b x h
Product
the answer to a multiplication problem
What is the reflection line of line segment ME if: M1 = (-5, 3) E1 = (-4, -4) And the new lines points are: M2 = (7, -1) E2 = (2, -6)
1 = Old 2 = New M1 = (-5, 3) M2 = (7, -1) E1 = (-4, -4) E2 = (2, -6) We need to find the midpoint(half way so divide by 2) between the difference of these two points: Point M: X: (-5 + 7) divided by 2 = 1 Y: (3 + -1) divided by 2 = 1 M = (1,1) Point E: X: (-4 + 2) divided by 2 = -1 Y: (-4 + -6) divided by 2 = -5 E = (-1,-5) The line that bisects between Line segments M1E1 and M2E2 equally is: LINE OF REFLECTION = A (1,1) to B(-1,-5)
How do you find the total number of triangles and angle degrees of an S sided polygon where: S = 102
102 sides To find the interior angles you subtract two sides: 102 - 2 x 180 = 18,000° for a 102 sided polygon
Plot the quadrilateral ABCD across the X axis where: A = (1,4) B = (-4,2) C = (-5,-4) D = (-2,-1)
A = (1,4) B = (-4,2) C = (-5,-4) D = (-2,-1) If we are reflecting then we are NOT rotating, we are FLIPPING! So positive X values become negative and Vice Versa, but Y values do NOT change. Therefore: A = (1,4) flipped 180 is (1,-4) B = (-4,2) flipped 180 is (-4, -2) C = (-5,-4) flipped 180 is (-5, 4) D = (-2,-1) flipped 180 is (-2, 1)
Plot the image of point A under a dilation about the origin (0,0) with a scale factor of 1/3): A = (3,-6) Dilation = (0,0) on the graph Scaled to: 1/3
A = (3,-6) Dilation = (0,0) on the graph Scaled to: 1/3 First, find relationship from origin to A: 3 to the right, -6 down. Then Scale it: divide by 3 to get (1,-3) A Scaled by 1/3 = (1,-3)
Trapezoid
A convex quadrilateral that has AT LEAST one pair of parallel sides.
Concave Quadrilateral
A four sided polygon that looks dented. Imagine an arrow which has four angles. One or more interior angles will be a Reflex Angle (over 180, but under 360).
Circle
A round plane figure whose boundary consists of points equidistant from the center.
What does it mean when a Circumcircle is Circumscribed about Δ ABC?
All three vertices lie on the circle and every point is on the circumradius of the circumcenter.
Geometric Translation
All vertex has shifted in the same direction by the same amount.
Reflex Angle
An angle (m∠ or just ∠) whose sum is greater than 180° BUT LESS THAN 360°. 𝟏𝟖𝟎° < (𝐑𝐞𝐟𝐥𝐞𝐱 𝐚𝐧𝐠𝐥𝐞 𝐦∠) < 𝟑𝟔𝟎°
Supplementary Angle
An angle (m∠ or just ∠) whose sum is 𝟏𝟖𝟎°.
Complete Angle
An angle (m∠ or just ∠) whose sum is 𝟑𝟔𝟎°.
Complementary Angle
An angle (m∠ or just ∠) whose sum is 𝟗𝟎°.
Subtended Angle
An angle who's two ends or vertices sit on the circumference of a circle and are connected by an arc. Refer to the image: If ∠ABC sits in a unique circle, and it's two ends are A and B with C being the subtended angle, then the Subtended angle will be 1/2 arc angle. Ex: For ∠AFB where F is the subtended angle, the angle between points A and B will be twice the amount of ∠F
Adjacent Angle
Angles that have a common side(line segment or ray) and a common vertex or third point. Ex: Complementary Angle: ∠ABC has two interior angles with a line segment BE. 𝐦∠𝐀𝐁𝐄 = 𝟒𝟎° 𝐚𝐧𝐝 𝐦∠𝐄𝐁𝐂 = 𝟓𝟎° So Line segments AB and BC are perpendicular because∠ABE ∠EBC share the point B and are adjacent.
Right Triangle
Any Side of a triangle that is the full diameter of the circumcircle is a Right Triangle.
Irregular Polygon
Any two dimensional shape who's interior angles and side lengths are congruent. (equal) equal angles + lengths
Polygon
Any two dimensional shape with only straight sides. Cannot be open or have curves.
Find : m∠MON: when: Ray OL ⊥ ON and: m∠LOM = 2X + 46° = m∠MON = 3X -6°
Find : m∠MON: Firstly, we know that Ray OL ⊥ ON so these adjacent angles (sharing 1 side) must be sum up to the complementary angle m∠LON = 90°. FIRST: Combine these two adjacent angle values knowing that when they are added together they MUST equal 90. Combine similar values and Isolate X: 2X + 46 + 3X - 6 = 90 5X + 40 = 90 5X.. = 50 ---- = ---- ..5 .......5 X = 10 PLUG INTO ∠MON Equation: m∠MON = 3X -6° m∠MON = 3(10) - 6 30 - 6 𝐦∠𝐌𝐎𝐍 = 𝟐𝟒° CHECK YOUR ANSWER: Check your answer by taking the value of X = 10 and plugging it into the angle value from m∠LOM = 2X + 46 2(10) + 46 20 + 46 = 66 m∠LOM = 66° 66° + 24° = 90° The answer m∠MON = 24° is correct!!!
Find m∠RPS: Where: Line QPS is supplementary (180°) And: m∠QPR = 2X + 122° = m∠RPS = 2X + 22°
Find m∠RPS: Firstly, we know that m∠QPS is 180° and that angles m∠QPR and m∠RPS are adjacent (next to one another) and when added will equal 180°. FIRST, combine values and then isolate X: 2X + 122 + 2X + 22 = 180 4X + 144 = 180 .......-144.. -144 ---------------- ............4X = 36 ............--- = --- .............4 .....4 X = 9 PLUG X INTO m∠RPS = 2X + 22° m∠RPS = 2(9) + 22 ....................... 18 + 22 𝐦∠𝐑𝐏𝐒 = 𝟒𝟎° CHECK BY PLUGGING IN m∠QPR = 2X + 122° m∠QPR = 2(9) + 122 .........................18 + 122 𝐦∠𝐐𝐏𝐑 = 𝟏𝟒𝟎° 𝟏𝟒𝟎° + 𝟒𝟎° = 𝟏𝟖𝟎° 𝐭𝐡𝐞𝐫𝐞𝐟𝐨𝐫𝐞 𝐭𝐡𝐞 𝐚𝐧𝐬𝐰𝐞𝐫 𝐢𝐬 𝐜𝐨𝐫𝐫𝐞𝐜𝐭!
Solve this translation on a number line when: (-169, 434) moves to (-203, -68) What are the coordinates of the image of point (31, -529) under this translation?
First, Find the difference: Point: (-169, 434) moves to (-203, -68) Therefore: -169 -X = -203 Trans. of X = -34 434 - y = -68 Trans. of Y : -502 Therefore: (31, -529) moves to: 31 - 34 = X value Corr. X Value is: -3 -529 -502 = -1031 SO: (31, -529) moves to: (-3, -1031)
Plot the image of Point A under a dilation about Point P with a scale factor of 3: "re-plot As relationship to B if it is scaled three times"
First, find the relation between Point A and P: (-2,-1) - to the left 2, down 1 Then scale it to 3x: (-6, -3) Finally: Recreate A in rescaled form to P: P = (0,0) A = (-2,-1) A-scaled = (-6,-3)
Find the interior angle measures of this irregular Polygon: You can do this by splitting the shape into two triangles.
First, split the polygon into two triangles: ΔABC and ΔXYZ = 180° A + B + C = 180° X + Y + Z = 180° A + B + C + X + Y + Z = 360°
Reflecting shapes using a diagonal line of reflection: y = -x - 2 and the slope is found by -x (-1/-1)
First: Find the reciprocate slope to find a perpendicular line: Slope now becomes 1:
Solve these angles of a RIGHT triangle: If MK ∥ NJ and ∠MKL and ∠NJL are both 90° what is m∠LMK? What is m∠LMK's relation to m∠LNJ? Triangle Angle = 180° total And we are given the values: A = ∠LNJ B = ∠LMK C = ∠NLJ
If MK ∥ NJ and ∠MKL and ∠NJL are both 90°: Triangle Angle = 180° total A = ∠LNJ B = ∠LMK C = ∠MLK First, express B as a formula: B + C + 90 = 180 B + C + 90 - 90 = 180 - 90 B + C = 90 𝐁 𝐨𝐫 ∠𝐋𝐌𝐊 = 𝟗𝟎° - 𝐂 Next, express A as a formula: A + C + 90 = 180 A + C + 90 - 90 = 180 - 90 A + C = 90 𝐀 𝐨𝐫 ∠𝐋𝐍𝐉 = 𝟗𝟎° - 𝐂 A = 90 - C = B 𝐀 = 𝐁 𝐀𝐧𝐠𝐥𝐞𝐬 𝐀 (𝐋𝐍𝐉) 𝐚𝐧𝐝 𝐁 (𝐋𝐌𝐊) 𝐚𝐫𝐞 𝐞𝐪𝐮𝐚𝐥.
Prove that this is a parallelogram: A convex quadrilateral that MUST HAVE two pairs of parallel lines and congruent angles. AB = DC AND AD = BC
If the corners are labelled from top left - A, top right- B, bottom right - C, and bottom left - D. If we draw a line between points BD and points AC we know we will have corresponding angles that are congruent. PROVE: AB ∥ DC ∠ABD ≅ ∠ BDC ∠DBC ≅ ∠ ADB And: Δ ADB ≅ Δ CBD = [ASA ≅] Therefore AD = BA (corresponding sides) congruent triangles. PROVE: AD ∥ BC Δ ACB≅ Δ DCB = [SSS ≅] And: ∠ABC ≅ ∠ DCB ∠ACB ≅ ∠ DBC
How many triangles can I fit inside of an S sided polygon?
If the first triangle takes 2 sides and then to make each other triangle takes 1 side, you do: S -2 = # of Δ
Point C is the image of (-4,-2) under a reflection across the Y axis. What is the original point of C:
If the reflection is (-4,-2) then it's reflection across the Y axis means it is a vertical line, therefore the X value will be the opposite of what it is. (4,-2) is the original point
What is the measure of these Vertical Angles: m∠BED = 7X + 182° and m∠CEA = 9X + 194°
If you have 2 intersecting lines AB and CD with intersecting point E: Vertical ∠ are represented as: m∠BED = 7X + 182° and it's reciprocal is m∠CEA = 9X + 194° FIRST isolate the variable X: 9X + 194 = 7X + 182 -7X -194 = -7X -194 ------------------------ ..............2X = -12 NEXT Further Isolate X: 2X x (1/2) = -12 x (1/2) Equals: X = -6 SO: 9(-6) + 194 = -54 -54 + 194 = 140° and 7(-6) + 182 = -42 -42 + 182 = 140° THEREFORE: ∠𝐁𝐄𝐃 = 𝟏𝟒𝟎° 𝐚𝐧𝐝 𝐦∠𝐂𝐄𝐀 = 𝟏𝟒𝟎°
Unique Circle
If you have 3 unique points that are NOT co-linear, and have a unique circumcenter and circumradius, you will define a unique circle.
Why is SSA not Postulate (true without proof).
If you have a triangle with two equal sides and one equal angle (Side, Side Angle) meaning that ONLY ONE side is part of the angle, and that is because we are assuming the length of one of the sides points outwards making a wider triangle, when really it could be the same length but make the 3rd (unknown side length) much smaller! UNLESS THE ANGLE IS OBTUSE!
Permutation Formula
Permutation Formula : P = Permutations x = # of people c = # of chairs ! = Factor symbol EQUAL VALUES: P(5)! = 5 x 4 x 3 x 2 x 1 = total UNEQUAL VALUES: For example 5 people, 3 chairs. ...............x! (people) P(x,c) = --------------- ............... (x - c)! (people - chairs) (𝟓 𝐱 𝟒 𝐱 𝟑 𝐱 𝟐 𝐱 𝟏)! 𝟓! 𝐏!(𝟓-𝟑)! ---------------------- = ------- . ............... (𝟐 𝐱 𝟏)!.....................𝟐!
How do you determine a Plane?
Planes are determined by finding 𝟑 𝐧𝐨𝐧 𝐜𝐨-𝐥𝐢𝐧𝐞𝐚𝐫 𝐩𝐨𝐢𝐧𝐭𝐬. Ex: A line that has Co-Linear points A, B, and C and non Co-linear point D. Existing planes could be determined by: ABD, ACD, BCD. You CANNOT say place ABC because they are co-linear which means they could sit on an infinite number of planes (imagine a bead being threaded by string (ABC).
Positive VS Negative Angle Rotations:
Rotate: To Rotate 60°" = +X° = Rotate LEFT To Rotate -60° = -X° = Rotate RIGHT
Arc Angle
The difference between two vertices or points on a circle's circumference measured by finding the circumcenter or center point of the circle. This is used to find the Subtended angle of another vertex or point.
What are the Permutations of a 3 letter word in the Alphabet using repeats? What about only not repeating letter?
There are 26 letters in the Alphabet. If you repeat letters you have: P(26)!=26 x 26 x 26 = 17,576 or = 26!/(26)! If you can't repeat the letters? (eliminate a letter for each space) 𝐏(𝟐𝟔)!= 𝟐𝟔𝐱𝟐𝟓𝐱𝟐𝟒 = 𝟏𝟓𝟔𝟎𝟎 or ............= 26!/(26-3)! 𝐏(𝟐𝟔)! = 𝟐𝟔! / 𝟐𝟑!
How do the angles made in AB and DEF crossed with transversal line GH relate to one another? What if ABC ∥ DEF ? What if ABC ∥ DEF and GH ⊥ ABC?
There are three lines, (ABC ∥ DEF), and (GH) is a line that intersects both ABC ∥ DEF. This means that: C & E can be the Vertex for Supplemental(180°) OR Complimentary angles (90°). C & E can ALSO be Complete Angles (360°) But we know that. ∠ABG = ∠DEB ∠GBC = ∠BEF ∠HEF = ∠EBC ∠ABE= ∠DEH ∠CBH = ∠FEH (𝐀𝐁𝐂) ∥ (𝐃𝐄𝐅) 𝐦𝐞𝐚𝐧𝐢𝐧𝐠 𝐭𝐡𝐚𝐭 𝐢𝐭'𝐬 𝐫𝐞𝐜𝐢𝐩𝐫𝐨𝐜𝐚𝐥 𝐚𝐧𝐠𝐥𝐞 𝐰𝐢𝐥𝐥 𝐛𝐞 𝐭𝐡𝐞 𝐬𝐚𝐦𝐞.
Using Permutations, how many variations of seating arrangements can be made? There is a table has 5 chairs, for 5 people to sit in. (obviously).
There is a table has 5 chairs, for 5 people to sit in. _____ ______ _____ _____ _____ ..1....... 2 .....3..... 4..... 5 -All 5 could possibly sit in 1st chair. -If 1st is taken, 4 left could sit in 2nd -If 2nd is taken, 3 could still sit in 3rd -If 3rd is taken, 2 can sit down on 4th -If 4th is taken, 1 can take the 5th. So: 𝟓 𝐱 𝟒 𝐱 𝟑 𝐱 𝟐 𝐱 𝟏 = 𝟏𝟐𝟎
Quadrilateral
Two dimensions shapes that have : 4 different sides 4 Vertices (points) 4 angles
Parallel Lines
Two lines that do not intersect and are co planar(two lines that share the same plane). These lines can also be translated on top of one another. Seen as: ∥
Perpendicular Lines
Two lines that intersect at one point and the angle they intersect at is a right angle, or 90°. Seen as = ⊥
Intersecting Lines
Two lines that intersect one another at a point but do not end at that common point. This is the difference between an intersecting point and a vertex.
Right Angle
Two rays that come from a common point or Vertex, or center (of a circle), who's interior angle is exactly 90°. Ex: Image a cookie who's complete angle is 360°, and the cookie is divided in four by being cut twice perpendicular to give you 4 equal pieces. Each piece from tip to edge of the cookie is equidistant and perpendicular making it 90°. 90° x 4 slices is 360° so the cut or line segments for each piece would be 90° exactly, making the angles of each quarter a right angle. *There will be a small box against the vertex of the angle if this is a true right angle.* 𝐑𝐢𝐠𝐡𝐭 = 𝟗𝟎° 𝐨𝐫 ∠𝐀𝐁𝐂 = 𝟗𝟎°
Obtuse Angle
Two rays that come from a common point or Vertex, or center (of a circle), who's interior angle is more than 90°. Ex: Imagine a tub of ice cream that has three flavors, chocolate, vanilla, and strawberry. Each flavor of ice cream takes an equal amount of space in the tub (1/3rd). If each flavor is 1/3 then 1/3 of 360° is 120°. The line segments connecting each flavor separation to the center where all three flavors meet creates an angle of 120°. This is comparable to when something or someone is dull, they are obtuse. 𝐎𝐛𝐭𝐮𝐬𝐞 > 𝟗𝟎° 𝐨𝐫 ∠𝐀𝐁𝐂 = 𝟏𝟐𝟎°∠𝐀𝐁𝐂 = 𝟏𝟐𝟎°
Find the radius of the circumcircle using the area of this Right Triangle: Use formula [ABC] = (0.5) b x h
We know: h = height b = base d = diameter cc = circumference [ABC] = area r = circumradius [ABC] = (0.5) b x h Therefore: 2(ABC) = 2( 1/2 x b x h) -------- = ---------------- ........b......................b The two's cancel out and the B cancels out: h = 2(ABC)/b or rewrite: c ....2[ABC] -- = ------- 2r .......ab Cross multiply and isolate r: abc = 4r[ABC] ---------------- ....4[ABC] Which gives you: r = abc/4[ABC]
Find the Circumcenter of a Right Triangle.
What do we know: We know that ∠OMB and ∠ABC are similar. ∠BC = 2(BM) ∠BA = 2(BO) 1 ....BM.....BO -- = --- = ---- 2......BC....BA Since these two are the same then that means OB = OC OB = OA and BM = BC Which means that: OC = OA this triangle is SAS Point O is equidistant to ALL vertices (points) on this triangle and the midpoint of a Right Triangle's Hypotenuse it is also the circumcenter. If a point sits on the perpendicular bisector of a segment. it will also be Equidistant to the center of the segment it intersects on the other line segment: AB. Because M is equidistant between BC and it is perpendicular, it will ALSO meet at point O on line segment AB. ∠C and ∠M are the same. Any Side of a triangle that is the full diameter of the circumcircle is a Right Triangle
Solve for X and Y given that two of the three angles are provided: ∠c = 64 ∠a = 50 + X ∠b = 31
What do we know: a + b + c = 180: Find the value of X: a(50 - X) + c(64) + b(31) = 180 50 + 64 + 31 = 145 180 - 145 = 35 X = 35 Therefore: ∠cab = (35) + 50 = 85 Find the Value of Y: a(50) + c (64) + Y = 180 114 + Y = 180 180 - 114 = 66 Y = 66
Solve for X in this geometric representation where: ∠ABC = 4x ∠DCE = 2x and lines AB ∥ CE Line segment AC is the transversal line
What we know: AB ∥ CE will have vertical angles (matching on opposite sides) because there is a transversal line (a line that intersects both parallel lines). So: ∠ABC = 4x so ∠FCD= 4x ∠DCE = 2x so ∠CBG = 2x Therefore: ∠ABC=4x + ∠CBG=2x = 180° Simplify: 4X + 2X = 180 6X = 180 ---------- ........6 X = 30 ∠ABC= 120° ∠DCE = 40°
Solve for the angles found in a polygon by using supplementary triangle angles. This 5 Sided uneven Polygon is labelled (as seen in the image). a, b, c, d, and e, for it's external supplementary angles. What angles are adjacent to these points to get 180°?
What we know: Because this polygon has exterior angle measurements, we know that it also must have Interior angles: f,g,h,i,j Match each exterior and interior angle to isolate the supplementary interior angles: must = 180 a = 180 - g b = 180 - h c = 180 - i d = 180 - j e = 180 - f 180 x 5 = 900 total angles of each line - sum of interior angles. 900 - (g + h + i + j + f) NOW: Make 3 triangles out of this polygon to further isolate the angles: f = k + l j = m + n h = o + p + q 900 (sum of exterior angles) 900° - (g+o+p+q+i+n+m+l+k)° ∠gko = 180° = T1 ∠plm = 180° = T2 ∠qin = 180° = T3 Therefore: Subtract the supplementary angles of the interior of each triangle from the total exterior angles and you get: 900-180(T1)-(T2)180-(T3)180 900 - 540 = 360°
What are the measures of a triangle where the Largest Angle is 4 times the second largest angle, and the smallest angle is 10°?
What we know: We know that the interior angles of any triangle must always equal 180° so: 180 = 4X + X + 10 Isolate the X variable: 170 = 5X ---------- .........5 X = 34° The triangle is: 4(34) + 34 + 10 = 180 136 + 34 +10 = 180°
Is this a true circumcenter of a triangle?
What we know: r = circumradius cc = circumcenter L + M = 180° this is a SAS Side angle side that has a Perpendicular Bisector AB = AE + EB BC = BF + FC AC = AG + GC Al segments (AB,CB,AC) can be perpendicularly bisected by the Circumcenter. If you take any circle, and put a triangle who's vertices sit on the circumference of it, you will be able to find the circumcenter.
Congruent
When the size or angle is the same in a shape or geometric representation. Seen as: ≅ or = With: Δ = triangle ⊥ = perpendicular ∠XYZ = angle measure X° = Angle Degree
RHS Postulate
When three sides of one triangle are congruent(same in length and angle) to three sides of another triangle, then the two triangles are congruent (the same).
Reflect the shapes using diagonal line: Y= -X - 2 Where
Y = -X - 2 is slope intercept form Slope: -1/-1 Y intercept: -2 If we want to reflect the point, we need to find the perpendicular slope: Perp slope of -1 is 1. Then we need to draw the perpendicular slope and count each point's distance and mirror that to the other side of the reflection line.
Looking geometric triangle provided: what is the value of ∠θ (Theta)? Y = 90° Z = 45° ∠1 = 32°
Y = 90 First Find X: 32 + 90 + X = 180 180-122 = X X = 58 Find the value of T: 32 + 90 + T = 180 180 - 122 = T T = 58 Find θ: 90+ 58 + θ = 180 180 - 148 = θ θ = 32
Geometric Rotation
You can rotate the shape by degrees via a single vertex or by the whole shapes orientation. So point C : ( 2,-6) becomes ( 6,-6). Or Orientation: The whole shape rotates betweeen the Y and X axies assuming that the center point is XY =00. IF YOU ROTATE -° YOU ROTATE CLOCKWISE. -90 = Clockwise 90 = CounterClockwise
What is a rigid transformation?
the whole shape goes through the same transformation, so with rotations, reflections, and translations, the shape should not change at all. You are NOT stretching or shrinking the image at all.
Translate: T(8,-1) ΔABC where A = (-1, 2) B = (-4, 8) C = (-7, 2)
ΔABC where A = (-1, 2) B = (-4, 8) C = (-7, 2) Translation: A = X(-1 + 8) Y(2 -1) = (7, 1) B = X(-4 + 8) Y(8-1) = (4, 7) C = X(-7 + 8) Y(2-1) = (1, 4)
If ΔPIN is rotates -270° about the origin, what way will the triangle rotate? P = (2,3) I = (7,-7) N = (2,-7)
ΔPIN is rotates -270°, therefore ΔPIN will rotate clockwise (to the right) 180° + an additional 90° So it will go 3/4 away around the graph to the "9 O'clock" position around the XY center point. -270° Rotation = If you start at 5,0 and rotate that -90 you get 0,-5, rotate that by another -90 and you get -5, 0, and rotate that by another -90 and you get 0,5. That makes a rotation of -270°. Therefore: P = (2,3) I = (7,-7) N = (2,-7) And: -270° = +90° So rotate all by +90° We do this by taking the points and bringing them to their corresponding Sqaure: So P = (2,3) Is in square TR or (positive x positive y) so we move it one to the left to make that the positive 90 degree rotation. P = (2,3) becomes (-3, 2) The 2X rotates to 2Y and the 3Y Rotates to becomes -3X. I = (7,-7) becomes: (7,7) The 7X value becomes 7Y value and the -7Y value rotates counterclockwise to become +7X N = (2,-7) becomes: (7, 2) The -7Y goes counter clockwise to become an X value that is positive and the 2X becomes the Y value that is STILL positive
Use a factor tree to find the factor of √-36.
√-36 V36 * √ -1 or i 36 = 2 *18 18 = 2 *9 9 = 3 * 3 √ 2 * 2 * 3 * 3 take pairs out from under the radicals sign. So 2 * 3 = 6. 𝟔𝐢 = √-𝟑𝟔
Principle Root
𝐎𝐍𝐋𝐘 𝐭𝐡𝐞 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 𝐢𝐧𝐭𝐞𝐠𝐞𝐫𝐬 𝐭𝐡𝐚𝐭 𝐟𝐚𝐜𝐭𝐨𝐫 𝐢𝐧𝐭𝐨 𝐭𝐡𝐞 𝐨𝐫𝐢𝐠𝐢𝐧𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫. You must change the negative Root in order to solve
Laws of Exponents
These Laws define all forms of Exponents where: M = Whole # Pt. or "Power" N = Fraction Pt or "Root" where X = 6, Y = 4, a = 3, b = 2) Law: Example: -where ^ is 1 X^1 = X = 6^1 = 𝟔^𝟏 -where ^ is 0 X^0 = 1 = 𝟔^𝟎 = 𝟏 -where ^ is -1 X^-1 = 𝟔^-𝟏 = -𝟏/𝟔 -where X * X with diff ^'s X^a*X^b = X ^(a+b) = 𝟔^𝟑 x 𝟔^𝟐 = 𝟔^(𝟑+𝟐) -where X/X with dif ^'s X^a/X^b= X^(a-b) = 𝟔^𝟑/𝟔^𝟐 = 𝟔^(𝟑-𝟐) -where parenthesis separates ^'s (X^b)^a = X^ba = (𝟔^𝟐)^𝟑 = 𝟔^𝟔 -where 2 variables affected by same ^. XY^b = X^b Y^b = (𝟔^𝟐)(𝟒^𝟐) -where X/Y are a fraction with ^. (X/Y)^b = 𝐗^𝐛/𝐘^𝐛 = 𝟔^𝟐/𝟒^𝟐 -where ^ is -negative. X^-b = 1/X^b = 𝟔^-𝟐 = 𝟏/𝟔^𝟐 -where a/b is an exponential fraction. X (a/b) = ^𝐛√𝐗^𝐚 = (^𝐛√𝐗)^𝐚
Find the probability of rolling doubles (6) and (6) together on two dice: What about any doubles? If you eliminate any doubles you have already rolled, what is the probability now?
These are all of the outcomes: Statistics: 1/36, or 2.7% .....[ 1 ]..... [ 2 ]..... [ 3 ]..... [ 4 ] .....[ 5 ] .....[ 6 ] [ 1 ] .....1, 1 ......1, 2 .....1, 3 ....1, 4...... 1, 5 .......1, 6 [ 2 ]....2, 1 .......2,2 .....2,3.....2,4....... 2,5...... 2,6 [ 3 ] ...3, 1...... 3,2 .....3,3 ....3,4 ......3,5 ......3,6 [ 4 ] ...4, 1 ......4,2 .....4,3 ....4,4 ......4,5 ......4,6 [ 5 ] ...5, 1 ......5,2 ......5,3 ....5,4 .....5,5....... 5,6 [ 6 ] ...6, 1 ......6,2 .....6,3 ......6,4..... 6,5...... 6,6 What about any doubles? 1/6 or 6/36 or 16.6~ If you eliminate any doubles you have already rolled, what is the probability now? First Roll: 𝟏𝟔.𝟔~ (𝟏/𝟔 𝐨𝐟 𝟏𝟎𝟎%) Second Roll: (double 6 removed) any double: 𝟓/𝟑𝟓 𝐨𝐫 𝟏𝟒.𝟐𝟖% (𝟏/𝟕 𝐨𝐟 𝟏𝟎𝟎) Specific Doubles: (𝟑'𝐬) 𝟏/𝟑𝟓 𝐨𝐫 𝟐.𝟖𝟓%
Distributive Property
Used with P.E.M.D.A.S. (Parenthesis, Exponents, Multiplication, Division, Addition, and Subtraction) 𝐰𝐡𝐞𝐧 𝐭𝐰𝐨 𝐭𝐞𝐫𝐦𝐬 𝐨𝐫 𝐯𝐚𝐫𝐢𝐚𝐛𝐥𝐞𝐬 𝐀𝐑𝐄 𝐍𝐎𝐓 𝐫𝐞𝐥𝐚𝐭𝐚𝐛𝐥𝐞. EX: 1/2(X-Y) - 4 = 1/2 (X) - 1/2 (Y) - 4. *you 𝐝𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐞 the 1/2 𝐭𝐨 𝐞𝐚𝐜𝐡 𝐗 𝐚𝐧𝐝 𝐘 even though the X and Y are in Parenthesis.
Calculating Compound Probability (Unfair): *Series of Independent Events*
An unfair Coin where Heads:Tails is 60:40 out of 100% Probability for each independent event twice in row: H = Outcome Heads T = Outcome Tails where: P(H) = 60%, 60:100, 6/10, or 3/5. and: P(T) = 40%, 40:100, 4/10 or 2/5 𝐏 (𝐇𝟏)(𝐇𝟐) = 𝐏 ( 𝟎.𝟔) 𝐱 𝐏 ( 𝟎.𝟔) = 𝟎.𝟑𝟔 What about P(T1,H2,T3) = 𝐏(.𝟒𝟎) 𝐱 𝐏(.𝟔𝟎) 𝐱 𝐏(.𝟒𝟎) = .𝟎𝟗𝟔 𝐨𝐫 𝟗.𝟔% *𝐭𝐚𝐤𝐞 𝐚𝐰𝐚𝐲 𝐞𝐚𝐜𝐡 𝐝𝐞𝐜𝐢𝐦𝐚𝐥 𝐛𝐮𝐭 𝐚𝐝𝐝 𝐀𝐋𝐋 𝐝𝐞𝐜𝐢𝐦𝐚𝐥𝐬 𝐛𝐚𝐜𝐤 𝐭𝐨 𝐭𝐡𝐞 𝐟𝐢𝐧𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫**
Rational Number
EX: 4/1 is a Real Number but not a Integer (whole number) -8/-2 is a Real Number and Integer 1/5 = .2 is a rational # but not an Integer.
Factor this quadratic using Common Factors: 4Y^2 + 4Y - 15 = 0 Group using G.C.F. (Greatest Common Factor)
Find variables A and B by: variable = sum X values to get 4Y constant = product of (4) x (-16) = -60 What two numbers give you -60: ( -1, 60) ( 1, -60) ( -2, 30) (2, -30) (-3, 20) (3, -20) (-4, 15) (4, -15) . (-6, 10) (6, -10) What two numbers summed give you 4: ( -6, 10) So: 4Y^2 + 4Y - 15 4Y^2 - 6Y + 10Y - 15 Group by GREATEST common factors: 2Y(2Y-3) 5(2Y - 3) Simplify the common factors so that: (𝟐𝐘 -𝟑) (𝟐𝐘 + 𝟓) = 𝟒𝐘^𝟐 + 𝟒𝐘 - 𝟏𝟓
Prove that ∠ABC ≅∠ACB is an isosceles triangle:
First, make two triangles by finding equidistant midpoint from both B and C. We know this because SSS and so their AAA.
Solve this Linear Equation: (-1/2) X + 3/4 = 5/6
(-1/2) X + 3/4 = 5/6 Convert all values to like terms and Isolate X: (-6/12)X + 9/12 = 10/12 ................ - 9/12 ....-9/12 ------------------------ -(1/2)X = 1/12 Get the reciprocal and multiply on both sides to further isolate X: (-1) x (-2) = 2. .........1 x (-2) = -2 ----------- -- so -------- ... --- (-2) x (-1) = 2........ 12 x (-1) = -12 ........................ 𝟐 𝐗 = - (𝐧𝐞𝐠) --- 𝐨𝐫 𝐗 = (-𝟏/-𝟔) ..................... -𝟏𝟐 Check your answer: (-1/2) (-1/6) + 3/4 = 5/6 -1 ..... -1 ..........1 ........9 ........10 ---- x ---- = --- + ---- = ----- = 5/6 -2....... -6 .......12 ......12........ 12 𝟓/𝟔 = 𝟓/𝟔 𝐒𝐨 𝐭𝐡𝐞 𝐚𝐧𝐬𝐰𝐞𝐫 𝐢𝐬 𝐜𝐨𝐫𝐫𝐞𝐜𝐭!
Factor the Quadratic: -X^3 + 17X^2 - 70X = 0
*Every term here is divisible by -X* Factor out a -X: -X^3 + 17X^2 - 70X = 0 -X(X^2 - 17X + 70) Now we have a standard quadratic equation: Product/Constant (multiply to get) = 70 Middle Co-Efficient/Variable (add to get) = -17 What two numbers produce 70: ( -1, -70) ( 1, 70) ( -2, -35) ( 2, 35) ( -5, -14) ( 5, 14) ( -10, -7) ( 7, 10). What two number sum to -17: ( -7, -10) So: -𝐗(𝐗-𝟏𝟎)(𝐗-𝟕) = -𝐗^𝟑 + 𝟏𝟕𝐗^𝟐 - 𝟕𝟎𝐗
Quadratic Formula
*To solve for the roots. x = -b ± √(b² - 4ac)/2a EX: "Solve for X if: a = 1 , b = 4 , c = -21 X^2 + 4X - 21 = 0 <===== Quadratic Equation ***USE FORMULA*** X = -(4) + or - √[(4)^2] - 4(1)(-21) / 2(1) X = -4 +/- √16 - 4 x 1 x (-21) / 2(1) X = -4 +/- √ 16 + (-4)*(1)*(-21) [- *- = +] X = -4 +/- √16 + 84 = 100 X = -4 +/- √ 100 / 2 X = -4 +/- 10 (divided by / ) 2 𝐗 = -𝟒 + 𝟏𝟎 / 𝟐 = 𝟑 -𝐨𝐫- 𝐗 = -𝟒 - 𝟏𝟎 / 𝟐 = -𝟕 *𝐰𝐞 𝐮𝐬𝐞 𝐭𝐡𝐢𝐬 𝐟𝐨𝐫 𝐩𝐫𝐨𝐛𝐥𝐞𝐦𝐬 𝐭𝐡𝐚𝐭 𝐚𝐫𝐞 𝐡𝐚𝐫𝐝 𝐭𝐨 𝐟𝐚𝐜𝐭𝐨𝐫!
Solve this Linear Inequality: -0.5X ≤ 7.5
-0.5X ≤ 7.5 Simplify the -0.5X and FLIP the sign: (-2)(-0.5) X ≥ 7.5 (-2) X ≥ -15 All X's will be Larger or Equal to -15. This Linear Expression can be written as (-𝟏𝟓, 𝐈𝐧𝐟𝐢𝐧𝐢𝐭𝐲)
Factor Negative Common Factors and Grouping: -12F^2 -38F + 22
-12F^2 -38F + 22 We can simplify the -12F by finding the GCF: -2(6F^2 + 19F - 11) Find Product of a x b (6) x (-11) = -66 ( -1, 66) ( 1, -66) ( -2, 33) ( 2, -33) ( -3, 22) ( 3, -22)( -6, 11) ( 6, -11). Find Sum factors equaling (a + b)= 19 -3 + 22 = 19 So: -2(6F^2 -3F + 22F -11) Find GCF to group: (6F^2 - 3F) (22F - 11) -2{ 3F(2F -1) x 11 ( 2F-1)} Combine the Common Factor 2F-1 : -2{(2F-1) (3F + 11)} -𝟏𝟐𝐅^𝟐 - 𝟑𝟖𝐅 + 𝟐𝟐 Factored is: -𝟐{(𝟐𝐅 - 𝟏)(𝟑𝐅 + 𝟏𝟏)}
Solve this Double Inequality: -16 ≤ 3X + 5 ≤ 20
-16 ≤ 3X + 5 ≤ 20 Isolate the X in one part of this compound inequality: -16 ≤ 3X + 5 ≤ 20 .-5 ...........-5 ....-5 _________________________ -21 ≤ 3X ≤ 15 Further Isolate by getting rid of X's Coefficient: -21 ≤ 3X ≤ 15 -------------------- ...........3 Therefore: -𝟕 ≤ 𝐗 ≤ 𝟑
Solve for this Inequality with Variables on both sides: -3P - 7 < P + 9
-3P - 7 < P + 9 Isolate the P: -3P -7 -3P < P + 9 +3P -9 +3P ........- 9 ----------- < -------- .............. -16 < ...4P Simplify: -16.... 4P --- < --- = -𝟒 < 𝐏 ...4 ......4 Check by plugging in a value into the original formula following the value that P must be bigger than -4.
Long Multiplication
Multiply the larger number by the smaller number starting with the first and then second number by each digit of the larger number.
How do you solve this simple linear equation: (-3)/4X = 10/13
Simplify to get the value of X: Simplify: -(3/4)X = 10/13 Multiply both sides by Co-efficient of X it's reciprocal. -(4/3) x -(3/4) = 10/13 x -(4/3) (10 x -4) / (13//3) 𝐗 = (-𝟒𝟎)/𝟑𝟗 Check to see if it is accurate: -3/4(-40)/39 = 10/13 simplify (-3 x -40) and (4 x 39) is: 120/ 156 which is all dividable by 4, so: 30/39 and divisible again by 3: 𝟑𝟎/𝟑𝟗 𝐝𝐢𝐯𝐢𝐝𝐞𝐝 𝐛𝐲 𝟑 𝐢𝐬 𝟏𝟎/𝟏𝟑 𝐬𝐨 𝐭𝐡𝐢𝐬 𝐚𝐧𝐬𝐰𝐞𝐫 𝐢𝐬 𝐜𝐨𝐫𝐫𝐞𝐜𝐭!
Improper Fraction
a fraction in which the numerator (top) is greater than or equal to the denominator (bottom). 𝐄𝐗: 𝟏 𝐚𝐧𝐝 𝟏/𝟒 = 𝟓/𝟒. *not commonly seen but good for determining which of two values with the same common denominator is greater/smaller.
Solve this Compound Inequality: 5X - 3 < 12 and 4X + 1 > 25
**This inequality doesn't have a solution** 5X - 3 < 12 and 4X + 1 > 25 Isolate both X's and Simplify: 5X - 3 < 12 and 4X + 1 > 25 .....+ 3 < +3 .................- 1 ...-1 ______________ .............._____________ ......5X < 15............... 4X > 24 Further Isolate the X's: 5X < 15 ........4X > 24 --------- ----------- 5 ........................4 Therefore: 𝐗 < 𝟑 𝐀𝐍𝐃 𝐗 > 𝟔 **𝐛𝐞𝐜𝐚𝐮𝐬𝐞 𝐢𝐭 𝐢𝐬 𝐚𝐧 𝐀𝐍𝐃 𝐚𝐧𝐝 𝐧𝐨𝐭 𝐚𝐧 𝐎𝐑 𝐩𝐫𝐨𝐛𝐥𝐞𝐦, 𝐭𝐡𝐞𝐫𝐞 𝐢𝐬 𝐧𝐨 𝐫𝐞𝐚𝐥 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧**
Factor this Quadratic as (X +/- a) (X +/- b): -X^2 - 5X + 24 = 0
-X^2 - 5X + 24 = 0 One X will be Negative and one will be positive: (-X + a) (X + b) You need to Factor the -1X^2 out so the equation then becomes: X^2 +5X -24 = 0 (X +/- a) (X +/- b) Factors of -24: (-1, 24) (-2, 12) (-3, 8) ( -4, 6) (-6, 4) ( -8, 3) ( -12, 2) (-24, 1) What factors added equal 5: (8, -3) So -1 (X - 3) (X + 8) = -X^2 + (-8X) + 3X + 24 -X^2 -5X + 24 = X^2 + 5x - 24 = -𝟏 (𝐗 - 𝟑) (𝐗 + 𝟖)
Find the Probability of flipping 2 heads on 3 coins: O = Heads X = Tails How many possible outcomes are there? Use a T diagram and a tree diagram to show.
1. X X X ...= 3X 2. X X O .= 2X ....1 O 3. X O X .= 2X ....1 O 4. X O O = 1X ....2 O 5. O X X ..= 2X... 1 O 6. O X O .= 1X ....2 O 7. O O X ..= 1X... 2 O 8. O O O .= 3 O Total Outcomes: 𝟖 Probability of flipping 2 heads on 3 coins one time: 𝟑/𝟖 Probability of flipping 2 heads on 3 coins after you get it once: 𝟐/𝟖 𝐨𝐫 𝟏/𝟒
Factor this Quadratic using Common Factors and Grouping: 35K^2 + 100K - 15
35K^2 + 100K - 15 These have a GCF of 5 so: 5(7K^2 + 20K - 3) Factors that give us (7[a] x -3[b] =-21 ( -1, 21) (-3, 7) ( 1, -21) ( 3, -7) Factors that sum (a + b) = 20 ( 21 , -1) = sum of 20 Put it into the equation: 5(7k^2 + 21K -1K - 3) Factor (find GCF) to group: 7K(K + 3) x -1(K + 3) so 5{7K(K + 3) -1(K + 3)} Simplify the K + 3: 𝟓(𝐊 + 𝟑)(𝟕𝐊-𝟏)
Solve for Y in this Compound Linear Inequality: "And" Both, only constrained by the inequality sign. 3Y + 7 < 2Y and 4Y + 8 > -48
3Y + 7 < 2Y and 4Y + 8 > -48 Isolate both Y's and Simplify: 3Y + 7 < 2Y and 4Y + 8 > -48 -3Y ........-3Y .................-8 ......-8 Then fully Isolate the Y's and FLIP when you Divide by a -number: 7 < -Y 4Y > ....-56 ------- .......---------- -1 ...........................4 𝐘 < -𝟕 𝐀𝐍𝐃 𝐘 > -𝟏𝟒 ** 𝐖𝐡𝐞𝐧 𝐠𝐫𝐚𝐩𝐡𝐢𝐧𝐠 𝐭𝐡𝐞 𝐥𝐢𝐧𝐞 𝐬𝐡𝐨𝐮𝐥𝐝 𝐛𝐞 𝐛𝐞𝐭𝐰𝐞𝐞𝐧 -𝟕 𝐚𝐧𝐝 -𝟏𝟒 𝐨𝐧𝐥𝐲.**
Factor this Quadratic Equation using formula: X = -b +/- √b^2-4ac / 2a <=== Equation 3x^2 + 6X = -10
3x^2 + 6X = -10 convert to: 3X^2 + 6X + 10 = 0 [a] [b] [c] where a = 3, b = 6, c = 10 X= -(6) +/- √ (6)^2 + (-4) x 3 x 10 / 2(3) X= -6 +/- √ 36 + (-120) / 6 𝐗= -𝟔 +/- √ (-𝟖𝟒) / 𝟔 *𝐘𝐨𝐮 𝐜𝐚𝐧'𝐭 𝐬𝐪𝐮𝐚𝐫𝐞 𝐚 𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞 𝐧𝐮𝐦𝐛𝐞𝐫, 𝐭𝐡𝐞𝐫𝐞𝐟𝐨𝐫𝐞 𝐭𝐡𝐞𝐫𝐞 𝐢𝐬 𝐧𝐨 𝐫𝐞𝐚𝐥 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧.*
Solve this Linear Equation to Graph: 4X - 3Y = 12 All Linear Equations form a Line
4X - 3Y = 12 Create a T - Chart to calculate where the line will be by 𝐟𝐢𝐧𝐝𝐢𝐧𝐠 𝐘 𝐚𝐧𝐝 𝐗 𝐈𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭𝐬: X Y Plug it in: 𝟎 , -𝟒 𝟑 , 𝟎
Factor by GROUPING this Quadratic: 4X^2 + 25X - 21
4X^2 + 25X - 21 variable = sum X values to get 25X constant = product of (4) x (-21) = -84 What two numbers give you -84 (product): (-1, 84) (-2, 42) (-3, 28) (-4, 21) (-6, 14) (-7, 12) (-12, 7) (-14, 6) (-21, 4) (-28, 3) (-42, 2) (-84, 1) What two numbers summed give you 25: (-3, 28) So: 4x^2 + 28X - 3X - 21 Terms are joined by common divisors: (4X^2 + 28X) + (-3X - 21) *divisor of 4 *divisor of -3 4X(X + 7) + -3(X + 7) Factor out X + 7 and simplify: (𝐗 + 𝟕)(𝟒𝐗 - 𝟑)
Calculating Compounding DEPENDENT Probability: There is a bag with 5 marbles, 2 being red and 3 being green. It is 35 Cents per try. 2 Green = 1$ What is the probability that your first marble would be green? What about your second? If you kept on playing what would your average earning be? Would you keep playing if it cost you 35 Cents per try?
5 Marbles total to start, 3 Green, 2 Red. Draw 1: 𝐏(𝐆𝟏) = 𝟔𝟎 / 𝟏𝟎𝟎 -if each GM is worth 20% each with all 5 marbles in the bag- ****You get it and so you take one of them out**** Draw 2: 𝐏(𝐆𝟏-𝟑- 𝐆𝟏) = 𝟓𝟎 / 𝟏𝟎𝟎 -If each GM is now worth 50% of all the marbles left in the bag.- So P(.60) x P(.50) = 30% Chance *Values are based on percentage* You would win roughly a dollar 30% of the times you play, earning you roughly 𝟑𝟎 𝐂𝐞𝐧𝐭𝐬 𝐚 𝐠𝐚𝐦𝐞 if you averaged each winning out. 𝐘𝐨𝐮 𝐰𝐨𝐮𝐥𝐝 𝐍𝐎𝐓 𝐰𝐚𝐧𝐭 𝐭𝐨 𝐤𝐞𝐞𝐩 𝐩𝐥𝐚𝐲𝐢𝐧𝐠 𝐭𝐡𝐞 𝐠𝐚𝐦𝐞 because when averaged, you are still losing 5 cents per game even if you win.
Solve for this Inequality with Parenthesis: 5X + 7 > 3(X + 1)
5X + 7 > 3(X + 1) Follow the parenthesis then Isolate the X: 5X + 7 > 3X + 3 -3X -7 > -3X -7 Further Isolate the X: 2X ..> -4 ---- > ---- ...2 ........2 Therefore: 𝐗 > -𝟐
Solve for Z in this Compound Linear Inequality: "OR" - Not in between, one or the other. 5Z + 7 < 27 or -3Z ≤ 18
5Z + 7 < 27 or -3Z ≤ 18 Isolate both Z's and Simplify: 5Z + 7 < 27 or -3Z ≤ 18 .......-7 < -7 .......----------- ..............................(-1/3) 5Z < 20 ≤ -18 -------- ≤ ----- 5 ................. 3 𝐅𝐋𝐈𝐏 𝐛/𝐜 -# 𝐝𝐢𝐯𝐢𝐝𝐞𝐝 𝐙 < 𝟒 𝐨𝐫 𝐙 ≥ -𝟔 **𝐰𝐡𝐞𝐧 𝐠𝐫𝐚𝐩𝐡𝐢𝐧𝐠. 𝐙 𝐢𝐬 𝐠𝐫𝐞𝐚𝐭𝐞𝐫 𝐭𝐡𝐚𝐧 𝟒 𝐎𝐑 𝐞𝐪𝐮𝐚𝐥 𝐭𝐨 𝐨𝐫 𝐥𝐞𝐬𝐬 𝐭𝐡𝐚𝐧 -𝟔.**
Factor by GROUPING this quadratic: 6X^2 + 7X + 1 = 0
6X^2 + 7X + 1 = 0 Middle Co-efficient A + B = sum of X values to get 7X Constant AB = product of (6) x (1) = 6 What two numbers give you product of 6: (-1, -6) (-2,-3) ( 2, 3) (1, 6) What two numbers summed (added) give you 7X: (1, 6) Join terms by common divisors: 6X^2 + 6X + 1X + 1 = (6X + 1) (X +1) 6X(X + 1) 1 (X+ 1) (6X + 1) (X + 1) Factor out X + 1 (𝐗 + 𝟏)(𝟔𝐗 +𝟏)
Solve this Linear Inequality: 75X ≥ 125
75X ≥ 125 Isolate the X: 75 X...... 125 ----- = ----- 75......... 75 So 𝐗 ≥ (𝟓/𝟑, 𝐈𝐧𝐟𝐢𝐧𝐢𝐭𝐲). All X's will be Larger or Equal to 5/3 or 1.33 or 1 2/3.
In statistics, what is a categorical variable?
A Categorical variable are 𝐰𝐚𝐲𝐬 𝐢𝐧 𝐰𝐡𝐢𝐜𝐡 𝐜𝐚𝐭𝐞𝐠𝐨𝐫𝐢𝐞𝐬 𝐨𝐟 𝐬𝐞𝐩𝐚𝐫𝐚𝐭𝐢𝐨𝐧 𝐨𝐫 𝐯𝐚𝐫𝐢𝐚𝐭𝐢𝐨𝐧 𝐚𝐫𝐞 𝐢𝐝𝐞𝐧𝐭𝐢𝐟𝐢𝐞𝐝 within the populous. For Example: Of 15 Coffee drinkers, 9 prefer their coffee hot, while the other 6 prefer it cold. The Type of Coffee: Hot or Cold is the Categorical Variable.
What is a fair compound probability with independent events? How do you calculate the compound events subsequently? Ex: flipping a coin 3 times to get Tails, Head, Tail
A fair compound probability with 𝐢𝐧𝐝𝐞𝐩𝐞𝐧𝐝𝐞𝐧𝐭 𝐞𝐯𝐞𝐧𝐭𝐬 𝐢𝐬 𝐚 𝐬𝐞𝐫𝐢𝐞𝐬 𝐨𝐟 𝐞𝐯𝐞𝐧𝐭𝐬 in which there is an 𝐞𝐪𝐮𝐚𝐥 𝐜𝐡𝐚𝐧𝐜𝐞 𝐨𝐟 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬, where each 𝐨𝐮𝐭𝐜𝐨𝐦𝐞 𝐝𝐨𝐞𝐬 𝐧𝐨𝐭 𝐚𝐟𝐟𝐞𝐜𝐭 𝐭𝐡𝐞 𝐧𝐞𝐱𝐭 𝐨𝐮𝐭𝐜𝐨𝐦𝐞. There is a 1 in 8 chance that you will get heads, tails and heads after that: Coin flip 1: 50:50 heads/tails = 1/2 Coin flip 2:50:50 heads/tails = 1/2 Coin flip 3:50:50 heads/tails = 1/2 So: 𝟏/𝟐 𝐱 𝟏/𝟐 = 𝟏/𝟒 𝐱 𝟏/𝟐 = 𝟏/𝟖
In statistics, what is a quantitative variable?
A quantitative variable is a 𝐦𝐞𝐚𝐬𝐮𝐫𝐞𝐦𝐞𝐧𝐭 𝐨𝐟 𝐞𝐚𝐜𝐡 𝐜𝐚𝐭𝐞𝐠𝐨𝐫𝐲 from a populous 𝐭𝐡𝐚𝐭 𝐜𝐚𝐧 𝐛𝐞 𝐢𝐝𝐞𝐧𝐭𝐢𝐟𝐢𝐞𝐝 𝐧𝐮𝐦𝐞𝐫𝐢𝐜𝐚𝐥𝐥𝐲 but cannot be otherwise categorized unless give more precise instruction. EX: Coffees have varying calories depending on what is ordered. Unless otherwise asked to classify based on hot or cold to separate the calorie count, there is no way to further categorize the calorie amounts provided in the data.
Mixed Number
A whole number (integer) with a fraction. 𝐄𝐗: 𝟏 𝐚𝐧𝐝 𝟏/𝟒 *𝐜𝐚𝐧 𝐛𝐞 𝐝𝐢𝐬𝐩𝐥𝐚𝐲𝐞𝐝 𝐚𝐬 𝐚𝐧 𝐢𝐦𝐩𝐫𝐨𝐩𝐞𝐫 𝐟𝐫𝐚𝐜𝐭𝐢𝐨𝐧. (𝟓/𝟒).*
What are the criteria for Triangle Postulates?
AAA = (angle angle angle) No, the triangle can be scaled. (same shape but NOT the same size) They are "Similar" UNLESS THEY SHARE THE SIDE SAS = (side angle side) Yes. If both sides and the angle in between are the same then there can ONLY be one length for the 3rd side! ASA = Yes, if both of the angles are congruent and they share a side, the other two sides MUST be the same! AAS = No, because the line between the angles can be different. Just because the length on one line is the same doesn't constrain the length of the other two sides just because they share the same angle. SSA = No, because while two sides of the triangle may be the same, the side C may be a different length and still share the same angle!
Real Numbers
All =/- numbers, whole(9), rational (20/9), and irrational (Pi) numbers that can be found on the number line. 𝐃𝐎𝐄𝐒 𝐍𝐎𝐓 𝐈𝐍𝐂𝐋𝐔𝐃𝐄 √ -𝟏 𝐨𝐫 𝐢𝐧𝐟𝐢𝐧𝐢𝐭𝐲
What is a simple Linear Equation?
An equation that can be graphed. The simple form is ax = b A is the variable co-efficient B is the constant. X is the variable.
Associative Property
As long as you follow the rules of P.E.M.D.A.S. (Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction) you can 𝐩𝐫𝐨𝐜𝐞𝐬𝐬 𝐩𝐚𝐫𝐭𝐬 𝐨𝐟 𝐚𝐧 𝐞𝐪𝐮𝐚𝐭𝐢𝐨𝐧 𝐢𝐧 𝐚𝐧𝐲 𝐨𝐫𝐝𝐞𝐫 𝐰𝐢𝐭𝐡𝐨𝐮𝐭 𝐚𝐟𝐟𝐞𝐜𝐭𝐢𝐧𝐠 𝐭𝐡𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞. EX: A(B*C) or B(A*C) This function will still produce the same outcome despite order.
Probability Formula *Compounding Events are Dependent*
Dependent events: where: P = Probability A = sequence or events you want to happen. O1 - # of Outcomes that satisfy A T1 - Total # of possible outcomes O2 - # of Outcomes that satisfy A once . O1 has been removed. T2 - Total # of possible outcomes after . . one of the total O1 has been . . removed. .............................𝐎𝟏....... 𝐎𝟐 𝐏(𝐀𝟏, 𝐀𝟐) = ------ 𝐱 ------ ...........................𝐓𝟏 .........𝐓𝟐 ex: Picking 2 Green Marbles Consecutively if you remove one of the green ones after taking it out.
Probability Formula (Fair): *Series of Independent Events*
Equally Likely Events: where: P = Probability A = sequence or events you want to happen. O - # of Outcomes that satisfy A T - Total # of possible outcomes ...............𝐎 (𝐛𝐞𝐬𝐭 𝐨𝐮𝐭𝐜𝐨𝐦𝐞) 𝐏(𝐀) = ------------------- ...............𝐓 (𝐭𝐨𝐭𝐚𝐥 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬) ex: dice P (even numbers) = 3//6 or 1/2
Common Factors
Factor (Pairs of integers) numbers can be used to find a common denominator in functions or find roots. (equation with input, variable, and output f(x)=x^2) EX: Find the greatest common factor of 32 and 22. 𝐅𝐚𝐜𝐭𝐨𝐫𝐬 𝐨𝐟 𝟑𝟐: 𝟏,𝟐, 𝟒,𝟖,𝟏𝟔,𝟑𝟐 𝐅𝐚𝐜𝐭𝐨𝐫𝐬 𝐨𝐟 𝟐𝟐: 𝟏,𝟐,𝟒, 𝟏𝟐, 𝟐𝟐) 𝐆𝐫𝐞𝐚𝐭𝐞𝐬𝐭 𝐜𝐨𝐦𝐦𝐨𝐧 𝐟𝐚𝐜𝐭𝐨𝐫 𝐢𝐬 𝟒.
Factor: (verb)
Find all: 𝐢𝐧𝐭𝐞𝐠𝐞𝐫𝐬 (𝐰𝐡𝐨𝐥𝐞 𝐧𝐮𝐦𝐛𝐞𝐫𝐬) 𝐭𝐡𝐚𝐭 𝐭𝐡𝐞 𝐨𝐫𝐢𝐠𝐢𝐧𝐚𝐥 𝐕𝐚𝐥𝐮𝐞 (𝐍𝐮𝐦𝐛𝐞𝐫) 𝐜𝐚𝐧 𝐛𝐞 𝐝𝐢𝐯𝐢𝐝𝐞𝐝 𝐛𝐲). EX: Factor 36: 1,2,3,4,6, 9, 12, 18,36.
Fractional Exponents
Fractions that indicate you should𝐟𝐢𝐧𝐝 𝐭𝐡𝐞 𝐬𝐪𝐮𝐚𝐫𝐞 𝐫𝐨𝐨𝐭 (^1/2) Square Root, cube (^1/3), or 4th root (^1/4) of a Value. X(^1/2) = √X X(^1/3) = 3√X X(^1/4) = 4√X
In probability, what is an independent event?
In an independent event, 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬 𝐩𝐫𝐢𝐨𝐫 𝐢𝐧 𝐧𝐨 𝐰𝐚𝐲 𝐞𝐟𝐟𝐞𝐜𝐭 𝐭𝐡𝐞 𝐰𝐚𝐲 𝐭𝐡𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬 𝐰𝐢𝐥𝐥 𝐛𝐞 𝐨𝐧 𝐲𝐨𝐮𝐫 𝐧𝐞𝐱𝐭 𝐭𝐮𝐫𝐧. Ex: A gambler winning the last 3 rolls is not inclined to win the next because all the rolls were independent events.
Irrational Numbers
Integers that CANNOT be written as a ratio (as a fraction) but are still a real number. EX: 𝐏𝐢 (𝟑.𝟏𝟒) 𝐜𝐚𝐧𝐧𝐨𝐭 𝐛𝐞 𝐰𝐫𝐢𝐭𝐭𝐞𝐧 𝐚𝐬 𝐚 𝐬𝐢𝐦𝐩𝐥𝐞 𝐟𝐫𝐚𝐜𝐭𝐢𝐨𝐧. 𝐓𝐡𝐞 √𝟐 𝐜𝐚𝐧𝐧𝐨𝐭 𝐛𝐞 𝐰𝐫𝐢𝐭𝐭𝐞𝐧 𝐚𝐬 𝐚 𝐟𝐫𝐚𝐜𝐭𝐢𝐨𝐧.
Solve this linear Equation where: X + 2X + 3 = -7X - 5
Isolate the Variable and it's Co-efficient. X + 2X + 3 = -7X - 5 3X + 3 = -7X - 5 Add 7X to both sides to get: 3X + 7X + 3 = - 5 10X + 3 = - 5 Subtract 3 from both sides to isolate the X. 10 X = - 8 Completely Isolate the X by dividing by the Co-efficient 10. 10X = - 8 ---..... --- ..10...... 10 𝐗 = (-𝟖/𝟏𝟎) 𝐨𝐫 (-𝟒/𝟓)
Solve this Linear Equation where: 5X - 3 - 7X = X + 8
Isolate the variable to one side: 5X - 3 - 7X = X + 8 Combine like variables on both sides: -2X -3 = X + 8 ...-X +3 = -X + 3 ------------------- -3X = 11 --------- -1/3 **Invert the -3 to -1/3 to get rid of the negative coefficient** Finish Isolating X: -3 X ......11 ----- = ----- -1/3..... -1/3 𝐗 = (-𝟏𝟏/𝟑) Check your Answer: 5(-11/3) - 3 - 7(-11/3) = (-11/3) + 8 -55/3 - 9/3 + 77/3 = (-11/3) + 24/3 -64/3 +77/3 = 13 /3 13/3 = 13/3 𝐗 𝐢𝐬 𝐜𝐨𝐫𝐫𝐞𝐜𝐭 𝐛𝐞𝐜𝐚𝐮𝐬𝐞 𝐛𝐨𝐭𝐡 𝐯𝐚𝐥𝐮𝐞𝐬 𝐚𝐫𝐞 𝐞𝐪𝐮𝐚𝐥 when plugging the value of X back into the equation!
Solve for this simple Linear Equation: (-5/6) X = 7/8
Simplify to get the value of X: Multiply both sides by the reciprocal: (-5)/6 X = 7/8 (-6)/5 x (-5)/6 and (-6)/5 x 7/8 This Isolates the X. (-6)/5 x 7/8 (-6) x 7 over 5 x 8 = 𝐗 = (-𝟒𝟐)/𝟒𝟎 𝐨𝐫 𝐗 = -𝟐𝟏/𝟐𝟎 Check your Answer: (-21)/20 x (-5)/6 (-21) x (-5) and (-20) x (-6) = 105 / 120 𝐒𝐢𝐦𝐩𝐥𝐢𝐟𝐢𝐞𝐝 𝐛𝐲 𝟏𝟓 𝐢𝐬: 𝟕/𝟖 𝐬𝐨 𝐭𝐡𝐞 𝐚𝐧𝐬𝐰𝐞𝐫 𝐢𝐬 𝐜𝐨𝐫𝐫𝐞𝐜𝐭!
Complete the table so each now represents a solution to the following equation: -3X + 7Y = 5X + 2Y Where: X Y -5 , ______ _____ , 8
Solve for the Missing X and Y variable that complete the solution using both points. -3X + 7Y = 5X + 2Y Where: ..X .....Y -5 , ______ _____ , 8 First, plug in (-5) to solve for Y: -3(-5) + 7Y = 5(-5) + 2Y Simplify: 15 + 7Y = -25 + 2Y Isolate Y: .. 15 + 7Y = -25 + 2Y +25 - 7Y = +25 - 7Y ----------------------- 40 = -5Y Simplify: -5Y ......40 ----- = ---- -5 ........-5 simplifies to: 𝐘 = -𝟖 𝐰𝐡𝐞𝐧 𝐗 = 𝟎 Now find the X value where Y = 8 -3X + 7(8) = 5X + 2(8) Simplify: -3X + 56 = 5X + 16 Isolate X: -3X + 56 = 5X + 16 +3X - 16 = +3X -16 --------------------- .........40 = 8X Simplify: 8 X ...40 --- = --- ..8....... 5 Simplifies to: 𝐗 = 𝟓 𝐰𝐡𝐞𝐧 𝐘 = 𝟖
Cummutatitive Property
States that you can change the order of values in problems involving addition or multiplication and you won't change the outcome. Ex: a+b=b+a or a*b = b*a 𝐃𝐎𝐄𝐒 𝐍𝐎𝐓 𝐀𝐏𝐏𝐋𝐘 𝐓𝐎 𝐒𝐔𝐁𝐓𝐑𝐀𝐂𝐓𝐈𝐎𝐍 𝐀𝐍𝐃 𝐃𝐈𝐕𝐈𝐒𝐈𝐎𝐍.
Venn Diagram
There are 12 Candies: 9 Chocolate, 3 Non-Chocolate. Of the 9 Chocolate, 3 are Coconut filled. Of the Non-Chocolate, 1 are coconut filled. So 1/3 of 9 Chocolate are coconut and 1/3 of Non-Chocolate are coated. nut. CH Only : 6 CH+CC : 3 NO CC+CC 1 NO CH or CC: 2
Two Way Frequency Tables:
There are 12 Candies: 9 Chocolate, 3 Non-Chocolate. Of the 9 Chocolate, 3 are Coconut filled. Of the Non-Chocolate, 1 are coconut filled. So 1/3 of 9 Chocolate are coconut and 1/3 of Non-Chocolate are coated. nut. Coconut No Coconut Has Choc: 3 6 No Choc: 1 2
Proportionality Test
Use cross multiplication to check whether or not two terms are proportional to one another even if the extremes are different. Top of Fraction = Means A C Bottom of Fraction = Extremes --- x --- B D A x D B x C *𝐩𝐫𝐨𝐩𝐨𝐫𝐭𝐢𝐨𝐧𝐚𝐥 𝐎𝐍𝐋𝐘 𝐢𝐟 𝐭𝐡𝐞𝐲 𝐚𝐫𝐞 𝐞𝐪𝐮𝐚𝐥.
What are Permutations and how can they be used?
Using factors to get the number of all possible outcomes, useful on larger numbers. Ex: A couch that sits 3 people( 1 2 3) and 3 people to sit in them (A , B, C) You could multiply or graph... 1 = A B C 2 = A C B 3 = B A C 4 = B C A 5 = C A B 6 = C B A ..........1.....2.....3 A = A1, A2, A3 B = B1, B2, B3 B = C1, C2, C3 BUT: You can also do it this way!!!! "If someone takes the first spot, how many are left to take the second, third, etc." ____ ____ ____ ...1.....2.....3 -All 3 can sit in Ch. 1 . -2 remaining can sit in Ch. 2 -Last 1 can sit in Ch. 3 3 x 2 x 1 = 𝟔 𝐏𝐨𝐬𝐬𝐢𝐛𝐢𝐥𝐢𝐭𝐢𝐞𝐬!
Solve for this Linear Inequality: X + 8 ≤ 6
X + 8 ≤ 6 Isolate the X, you do NOT have to flip the inequality sign: X +8 -8 ≤ 6 - 8 𝐗 ≤ -𝟐
What is the slope of the Line that contains these points: X - 2 3 4 5 Y - 4 1 -2 -5
X ...- 2 ...3.... 4 ...5 Y ...- 4 ....1 -..2 ..-5 Our slope is Rise over Run so When X = 2 and Y = 4 and our X changes to 3 and our Y changes to 1 we know that: Change in Y = 4 + ____ = 1 Change is.............. -3 Change in X = 2 + ___ = 3 Change is............... +1 Therefore Slope = -3 Rise over 1 Run 𝐒𝐥𝐨𝐩𝐞 = -𝟑
Find the Y-Intercept (where X=0) using the table: X Y -2 8 1 2 2 0 4 -4
X ....Y -2... 8 1 .....2 2.... 0 4...-4 Now add when X= 0 to the table X.... Y -2... 8 -1... ___ 0 ...___ 1 .....2 2 ....0 4 ..-4 When X is 1 and Y is 2, and X becomes 2 and Y becomes O that means that: ................0 - 2 (y2-y1) ..........-2 Slope = --------------- = -------- .................2 - 1 (x2-x1) .............1 Slope = -2 So 𝐰𝐡𝐞𝐧 𝐗 = 𝟎, 𝐘 𝐰𝐢𝐥𝐥 𝐛𝐞 𝐚𝐭 𝟒 because of the slope -(2)/1. 𝐄𝐯𝐞𝐫𝐲 𝐭𝐢𝐦𝐞 𝐭𝐡𝐞 𝐗 𝐚𝐱𝐢𝐬 𝐠𝐨𝐞𝐬 𝐮𝐩 𝟏 𝐩𝐨𝐢𝐧𝐭 𝐭𝐡𝐞 𝐘 𝐚𝐱𝐢𝐬 𝐰𝐢𝐥𝐥 𝐝𝐞𝐜𝐫𝐞𝐚𝐬𝐞 𝐛𝐲 𝟐 𝐩𝐨𝐢𝐧𝐭𝐬.
Factor this Quadratic Equation using formula: X = -b +/- √b^2-4ac / 2a <=== Equation -3X^2 + 12X + 1 = 0
X = -3x^2 + 12x + 1 = 0 where a = -3, b = 12, c = 1 X = -(12) +/- √(12)^2 + (-4) x (-3) x 1 / 2(-3) X = -12 +/- √144 + (-4) x (-3) x 1 / -6 X = -12 +/- √144 + 12 / -6 X = -12 +/- √156 / (-6) Simplify: √156 156 = 2 x 78 78 = 2 x 39 so... √156 = √2 x 2 x 39 = √2 x 2 (x) √39 = 2√39 therefore.... X = -12 +/- 2√39 / (-6) To get rid of the 2√ you divide everything by 2. X = -6 +/- √ 39 ... ------... ------ ....... -3 .........-3 X = 2 +/- √39 .................. ---- ................... -3 So: 𝐗 = 𝟐 + √𝟑𝟗/𝟑 𝐚𝐧𝐝 𝐗 = 𝟐 - √𝟑𝟗/𝟑
It is a Linear Equation if:
X = 3Y - 8 Y = 3^2 +6X - 1 3Y = 2^2 - 14 𝐄𝐯𝐞𝐫𝐲 𝐭𝐞𝐫𝐦 𝐢𝐬 𝐞𝐢𝐭𝐡𝐞𝐫 𝐚 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝐨𝐫 𝐚 𝐧𝐮𝐦𝐛𝐞𝐫 𝐭𝐢𝐦𝐞𝐬 𝐗 𝐨𝐫 𝐘 𝐚𝐧𝐝 𝐲𝐨𝐮 𝐚𝐫𝐞𝐧'𝐭 𝐦𝐮𝐥𝐭𝐢𝐩𝐥𝐲𝐢𝐧𝐠 𝐲𝐨𝐮𝐫 𝐗'𝐬 𝐚𝐧𝐝 𝐘'𝐬 𝐭𝐨𝐠𝐞𝐭𝐡𝐞𝐫, then it IS a linear equation!
Find the Intercepts of the line from this equation: -5X + 4Y = 20
X intercept: When Y = 0 Y intercept: When X =0 -5X + 4Y = 20 Plug in each variable as 0 to find the intercepts: -5X + 4(0) = 20 Isolate the X: -5X .....20 ---- = ---- -5 ......-5 X Intercept is: -4 -5(0) + 4Y = 20 Isolate the Y: 4 Y...... 20 ---- = ----- 4 .........4 𝐗 𝐈𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭 𝐢𝐬: 𝟓 𝐒𝐨 ( -𝟒, 𝟎) 𝐚𝐧𝐝 (𝟎 , 𝟓) Now you can graph having found the X and Y intercepts!
Solve this Linear Inequality: X/(-15) < 8
X/(-15) < 8 Isolate the X by multiplying both sides by -15 and FLIP the inequality sign: X ..................8(-15) --------- < --------- (-15) (-15) .......1 𝐗 > (-𝟏𝟐𝟎) 𝐒𝐨 (-𝟏𝟐𝟎,𝐈𝐧𝐟𝐢𝐧𝐢𝐭𝐲) 𝐓𝐡𝐞 > 𝐬𝐢𝐠𝐧 𝐝𝐨𝐞𝐬 𝐧𝐨𝐭 𝐢𝐧𝐜𝐥𝐮𝐝𝐞 𝐭𝐡𝐞 𝐯𝐚𝐥𝐮𝐞 𝐨𝐟 -𝟏𝟐𝟎
Solve this Linear Inequality: X/(-3) > (-10)/9
X/(-3) > (-10)/9 Multiply -3/1 to both sides to invert, and FLIP the inequality sign: (-3)(-1/3) X < (-10/9)(-3) This Isolates X and then we Multiply the Numerator and Denominator by 3: **𝐖𝐡𝐞𝐧 𝐦𝐮𝐥𝐭𝐢𝐩𝐥𝐲𝐢𝐧𝐠 𝐚 𝐝𝐞𝐧𝐨𝐦𝐢𝐧𝐚𝐭𝐨𝐫 𝐛𝐲 𝐚 𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞, 𝐲𝐨𝐮 𝐚𝐜𝐭𝐮𝐚𝐥𝐥𝐲 𝐝𝐢𝐯𝐢𝐝𝐞** X < -30/9 Divided by -3 is 10/3 𝐗 < 𝟏𝟎/𝟑 𝐨𝐫 𝟑.𝟑𝟑𝟑 𝐨𝐫 𝟑 𝟏/𝟑
Factor Quadratic (Quadratic Polynomial/Expression)
X^2 + 10x + 9 = 0 We want to find the value of A and B. (X + a) x (X + b) X^2 + bX + aX + ab X^2 + (a+b)X + ab X^2 + [(a+b) = 10] + [(ab) = 9] What are the factors of 9: 1,3,9 1+9= 10 a =1 b=9 (𝐗 + 𝟏) (𝐗 + 𝟗) 𝐚𝐫𝐞 𝐭𝐡𝐞 𝐟𝐚𝐜𝐭𝐨𝐫𝐬!
Factor this Quadratic as (X + a)(X + b): X^2 + 15X + 50 = 0
X^2 + 15X + 50 = 0 (X + a) x (X + b) What are the factors of 50: 1, 2, 5, 10, 25, 50 Factors that when added make 15: 5, 10 So (X + 5) x (X + 10) = X^2 + 10X + 5X + 50 = 0 𝐗^𝟐 + 𝟏𝟓𝐗 + 𝟓 = (𝐗 + 𝟓)(𝐗 +𝟏𝟎)
Factor this Quadratic as (X +/- a)(X +/- b): X^2 + 5X -14 = 0
X^2 + 5X - 14 = 0 (X +/- a) (X +/- b) What numbers factor into -14: 1, 2, 7, 14, -1, -2, -7, -14 What numbers add to make 5: -2, 7. Because the 5X is positive, the value of the negative b must be lesser than the value of positive a to equal a positive aX product (X-2)(X+7) = X^2 + 5X - 14
Factor this Quadratic as (X + a)(X + b): X^2 - 11X +24 = 0
X^2 -11X + 24 = 0 (X +/- a) (X +/- b) What numbers factor into 24: 1, 2, 3, 4, 6, 8, 12, 24 What Numbers added make - 11: -12, 1 but that can't be it because when multiplied it doesn't equal +24. BUT -3 and -8 when multiplied are factors of +24 and when added = -11 (X-3) x (X-8) do it out to check: X^2 + (-8X) + (-3X) + 24 = 0 𝐗^𝟐 -𝟏𝟏𝐗 +𝟐𝟒 = (𝐗-𝟑)(𝐗-𝟖)
Linear Slope
Y = Rise/vertical X = Run/horizontal 𝐒𝐥𝐨𝐩𝐞 = 𝐑𝐢𝐬𝐞 𝐨𝐯𝐞𝐫 𝐫𝐮𝐧 The higher the slope, the steeper the inclination of the line. The direction of points does not change the slope.
It is NOT a Linear Equation if:
Y = X^2 XY = 12 5/X + Y = 10 term-58 None of these are linear Equations because 𝐗 𝐜𝐚𝐧'𝐭 𝐛𝐞 𝐭𝐡𝐞 𝐝𝐞𝐧𝐨𝐦𝐢𝐧𝐚𝐭𝐨𝐫, We 𝐜𝐚𝐧𝐧𝐨𝐭 𝐝𝐢𝐬𝐜𝐞𝐫𝐧 𝐭𝐡𝐞 𝐯𝐚𝐥𝐮𝐞 𝐨𝐟 𝐗 𝐨𝐫 𝐘 by plugging in a number to find the values. If Every term is either a constant or a number times X or Y and you aren't multiplying your X's and Y's together, then it IS a linear equation!
How do you simplify aX^2 to factor the quadratic?
You find what 𝐠𝐢𝐯𝐞 𝐲𝐨𝐮 𝐭𝐡𝐞 𝐬𝐮𝐦 𝐚𝐧𝐝 𝐩𝐫𝐨𝐝𝐮𝐜𝐭 𝐨𝐟 𝐭𝐡𝐞 𝐁 𝐚𝐧𝐝 𝐂 values in the equation and then group common divisors to simplify. Breakdown: (FX + G) x (HX + J) FHX + FJX + GHX + GJ add the middle terms: (FH)X + (FJ +GJ)X + (GJ) A = FJ B = GH A + B = Middle Co-efficient. A x B = FJ x GH = First Co-efficient x Constant term. 𝐓𝐡𝐢𝐬 𝐢𝐬 𝐡𝐨𝐰 𝐰𝐞 𝐜𝐚𝐧 𝐟𝐢𝐧𝐝 𝐨𝐮𝐭 𝐰𝐡𝐚𝐭 𝐀 𝐚𝐧𝐝 𝐁 𝐯𝐚𝐥𝐮𝐞𝐬 𝐞𝐯𝐞𝐧 𝐚𝐫𝐞.
Calculating Combined (Fair and Unfair) Coin flip: *Dependent but NOT compounding Events* You have 8 Coins in a bag, 3 of them are Unfair (60:40)P heads over tails. 5 Are Fair (50:50)P Heads over tails. You randomly choose one coin and flip it two times. What is the percent probability of getting two heads?
You have 8 Coins in a bag, 3 of them are Unfair (60:40)P heads over tails. 5 Are Fair (50:50)P Heads over tails. You randomly choose one coin and flip it two times. F = 5/8 or 62.5% Fair U = 3/8 or 37.5% Unfair 0 = 5:10 or 50% Head θ = 5:10 or 50% Tails O = 6:10 or 60% Head ⊕ = 4:10 or 60% Tails FAIR 5:5 P(0 x 0 x F} = .50 x .50 x .625 = .15625 or 15.625% UNFAIR 6:4 P(O x O x U) = .60 x .60 x .375 = .135 or 13.5% TOTAL CHANCE: 15.625% on a Fair Coin 13.5% on an Unfair Coin 29.125% Combined 𝟐𝟗% 𝐂𝐡𝐚𝐧𝐜𝐞 𝐨𝐟 𝐠𝐞𝐭𝐭𝐢𝐧𝐠 𝟐 𝐇𝐞𝐚𝐝𝐬 𝐨𝐧 𝐚𝐧𝐲 𝐜𝐨𝐢𝐧.
Sum
the answer to an addition problem
