Math #2 Exam
Solve the system by substitution. If the system is inconsistent or has dependent equations, say so. x=10y+21 x=4/5y
(-3, -12/5)
Find the X and Y intercept for 4/3x - 4y= 3
(0,-3/4) = y intercept and (3,0) = x intercept
-1/5
1 unit down and 5 to the right
standard form: y + 4= -2(x-7)
2x+y=10
Decide whether the given lines are parallel, perpendicular, or neither. Remember The lines are perpendicular if the product of their slopes is minus−1. This means that the slopes are negative reciprocals.
3x-y=7 x+3y=-5
Write an equation in standard form of the line passing through the points ((12, 4) and (−3, 9). Remember to first find the slope and then use point slope formula.
3y+x=24
standard form ( A and B cannot be zero, and fractions are not allowed) make it positive by multiplying by negative 1
Ax+By=C
Solve the system by substitution. If the system is inconsistent or has dependent equations, say so. y=6x 18x−3y=0
If both variables are eliminated when a system of linear equations is solved, then there are two possible answers to the system of equations. 1. There are infinitely many solutions if the resulting statement is true. 2. There is no solution if the resulting statement is false. Since 0equals=0 is a true statement, there are infinitely many solutions. The system is consistent with dependent equations. Since the equations define the same line, the solution set is the set of all ordered pairs that satisfy either equation. The system has dependent equations and an infinite number of solutions.
When solving an inequality
If the graphs of the equations are the same, then there are an infinite number of solutions that are true for both equations.
When solving an inequality
If the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.
midpoint formula
M=(x1+x2/2, y1+y2/2)
Segment PQ has the given coordinates for one endpoint P and for its midpoint M. P= (-7,7) M= (-3,5)
Q=(1,3)
Graph the inequality. x+3≥0
Rewrite it as x+0y≥-3 and solve.
Pure acid is to be added to a 10% acid solution to obtain 18 L of a 60% acid solution. What amounts of each should be used?
The desired mixture will require 10 L of pure acid and 8 L of 10% solution.
How many gallons each of 6% alcohol and 10% alcohol should be mixed to obtain 240 gal of 8% alcohol?
The mixture contains 120 gal of 6% alcohol and 120 gal of 10% alcohol. These values check in the original problem.
A baseball team has home games on Friday and Sunday. The two games together earn $3177.50 for the team. Friday's game generates $278.50 less than Sunday's game. How much money was taken in at each game?
The team earned $1449.50plus+$1728.00equals=$3177.50 from both games. Since $1728.00minus−$1449.50equals=$278.50, the team earned $278.50 less from Friday's game than Sunday's game.
Write each equation in slope-intercept form and then tell how many solutions the system has. Do not actually solve. 2x+11y=10 6x+33y=4
They have no solutions because their slopes are parallel
Write an equation of the line that satisfies the given conditions. Give the equation (a) in slope-intercept form and (b) in standard form. m=−3, (2,4)
a) y=-3x+10 b)3x+y=10
If a container of liquid contains 80 oz of solution, what is the number of ounces of pure acid if the given solution contains the following acid concentrations? (a) 30% (b) 35% (c) 50% (d) 70%
a. 24 b.28 c. 40 d. 56
Write an equation of the line passing through the given point and satisfying the given condition. Give the equation (a) in slope-intercept form and (b) in standard form. (30, 8); perpendicular to 6x−y =3
a. y=- 1/6 x + 13 b. x+6y=78
Write an equation of the line passing through the given point and having the given slope. Give the equation (a) in slope-intercept form and (b) in standard form. (3,13.25) slope 3.75 Remember slope is same as m. if you need to get rid of decimals which is the same as a fraction, multiply everything by another fraction to get a whole number.
a. y=13.75x+2 b. 15y-4x=-8
Write an equation of the line passing through the given point and satisfying the given condition. Give the equation (a) in slope-intercept form and (b) in standard form. (9, 11); parallel to 8x−y=2 Remember 8x is the slope
a. y=8x-71 b. 8x-y=71
Write an equation of the line that satisfies the given conditions. Give the equation (a) in slope-intercept form and (b) in standard form. Through (−4,9); perpendicular to x=3
a. y=9 b. y=9
or
as long as the point works for one of the equalities, you can shade the region.
Solve the system by elimination. If the system is inconsistent or has dependent equations, say so. 11x+11y=40 x+y=55 Notice that 0equals=negative 15−15 is not a true statement. The system of equations is inconsistent and the solution is the empty set if the above process leads to a false statement, and consistent (equations are independent) if it leads to a unique solution. The system has an infinite number of solutions if it leads to a statement that is always true. The equations of such systems are dependent. Since the system is inconsistent, the solution set is the empty set.
empty set... no solution
y=6
horizontal slope
Negative slope lines fall from
left to right
Positive slope lines rise from
left to right
An official playing field (including end zones) for the Indoor Football League has a length 32 yd longer than its width. The perimeter of the rectangular field is 168 yd. Find the length and width of the field.
length is 58 yards width is 26 yards
a line under > means solid line, and no one means dashed
remember!
if the test point is true,
shade on that side
Find the slope of the line, and sketch its graph. y=5
slope= 0
Graph the intersection or union, as appropriate, of the solutions of the pair of linear inequalities. x+y≤22 and x≥1
step 1. plot the points step 2. pick points from each of the 4 regions step 3: test the points in both inequalities. shade the region that works for both inequalities.
Solve the system by elimination. If the system is inconsistent or has dependent equations, say so. 5x+6y=−4 3x+4y=−4
step 1: Multiply 5x + 6y = -4 by 4 step 2: Multiply 3x +4y=-4 by 6 step 3: add step four: solve for x and then substitute it back into either equation for y
Graph the linear inequality. 5x−y < 5
step 1: find two points step 2: plot the points step 3: find a test point and shade
In order to solve a system of linear equations
step 1: plot points step 2: find the point that makes lines intersect step 3: test the point into the original equations to see if its true
Find the slope of the following line, and sketch the graph. y=3x
the slope is 3
A straight line on a graph has a slope that is
undefined or 0
a slope that is 5
up 5 units and over 1 unit
For solving systems of linear equations in 3 variables,
use substitution method if all equations do not have x, y, z
x=4
vertical slope
Find the x and y intercepts x=-11
x intercept = (-11,0) has no y intercept
Write an equation in standard form of the line passing through (6,5) with an undefined slope.
x=6
point slope form
y-y1=m(x-x1)
How to find the slope
y2-y1/x2-x1 or rise over run
finding the slope of a graphed line
y= change in y/change in x times x plus the intercept on the y axis
Write 2x+3y=6 in slope intercept form
y=-2/3x +2
Write an equation in standard form of a horizontal line passing through (negative −5,5).
y=5
slope intercept
y=mx+b
