ALGEBRA
Linear Equations in Two Variables A linear equation in two variables, x and y, can be written in the form ax + by = c where a, b, and c are real numbers and neither a nor b is equal to 0. For example, 3x + 2y = 8 is a linear equation in two variables
Solving Quadratic Equations A quadratic equation in the variable x is an equation that can be written in the form ax^2 +bx^2 + c= 0 a, b, and c are real numbers and a ≠ 0 . Quadratic equations have zero, one, or two real solutions. The Quadratic Formula One way to find solutions of a quadratic equation is to use the quadratic formula: -b ± sqr root b^2 - 4ac x = -------------------------------- 2a Some quadratic equations can be solved more quickly by factoring
The Distance Between Two Points d=√((x_2 - x_1)²+(y_2 - y_1)²)
Mid point formula =(x1 +x2/2 , y1 +y2/2)
The Unknown Multiplier is particularly useful with three-part ratios: A recipe calls for amounts of lemon juice, wine, and water in the ratio of 2 : 5 : 7. If all three combined yield 35 milliliters of liquid, how much wine was included? Lemon Juice + Wine + Water = Total 2x + 5x + 7x = 14x Now solve for x: 14x = 35, or x = 2.5. Thus, the amount of wine is 5x = 5(2.5) = 12.5 milliliters. IMPT You can only use the Unknown Multiplier only once per problem to solve though. So if the dogs to cats is 2 : 3 and cats to mice is 5 : 4, you shouldn't write 2x : 3x and 5y : 4y. Instead, you need to make a common term.
Multiple Ratios: Make a Common Term You may encounter two ratios containing a common element. To combine the ratios, you can use a process similar to creating a common denominator for fractions. In a box containing action figures of the three Fates from Greek mythology, there are three figures of Clotho for every two figures of Atropos, and five figures of Clotho for every four figures of Lachesis. What is the least number of action figures that could be in the box? What is the ratio of Lachesis figures to Atropos figures? a) In symbols, this problem tells you that C : A = 3 : 2 and C : L = 5 : 4. You cannot instantly combine these ratios into a single ratio of all three quantities, because the terms for C are different. However, you can fix that problem by multiplying each ratio by the right number, making both C's into the least common multiple of the current values: C : A : L C : A : L 3 : 2 → Multiply by 5 → 15 : 10 5 : : 4 → Multiply by 3 → 15 : : 12 This is the combined ratio: 15 : 10 : 12 The actual numbers of action figures are these three numbers times an Unknown Multiplier, which must be a positive integer. Using the smallest possible multiplier, 1, there are 15 + 12 + 10 = 37 action figures b) Once you have combined the ratios, you can extract the numbers corresponding to the quantities in question and disregard the others: L:A = 12:10, which reduces to 6:5.
piecewise-defined function h(x) = {x, x_>_ 0 , - x, x<0 } The graph of this function is V-shaped and consists of two linear pieces, y = x and y = −x, joined at the origin In general, for any function h(x) and any positive number c, the following are true. The graph of h(x) + c is the graph of h(x) shifted upward by c units. The graph of h(x) − c is the graph of h(x) shifted downward by c units. The graph of h(x + c) is the graph of h(x) shifted to the left by c units. The graph of h(x − c) is the graph of h(x) shifted to the right by c units In general, for any function h(x) and any positive number c, the following are true. The graph of ch(x) is the graph of h(x) stretched vertically by a factor of c if c > 1. The graph of ch(x) is the graph of h(x) shrunk vertically by a factor of c if 0 < c < 1 .
RATIOS If two quantities have a constant ratio, they are directly proportional to each other. For example: If the ratio of men to women in the office is 3 : 4 there could be 3 men and 4 women, 9 men and 12 women, or even 600 men and 800 women, but there could not be 4 men and 3 women because then the number of men divided by the number of women would NOT equal ½ Label Each Part of the Ratio with Units The order in which a ratio is given is vital
Rates & Work Rate × Time = Distance OR Rate × Time = Work Basic Motion: The RTD Chart If a car is traveling at 30 miles per hour, how long does it take to travel 75 miles? Rate (miles/hr)× Time (hr)= Distance (miles) Car 30 mi/hr × = 75 mi 30t = 75, or t = 2.5 hours
Rates & Work Matching Units in the RTD Chart All the units in your RTD chart must match up with one another. The two units in the rate should match up with the unit of time and the unit of distance. For example: It takes an elevator four seconds to go up one floor. How many floors will the elevator rise in two minutes?9 check the units in the problem The rate is 1 floor/4 seconds, which simplifies to 0.25 floors/second. Note: The rate is NOT 4 seconds per floor! This is an extremely frequent error. Always express rates as "distance over time," not as "time over distance." IMPT The time is 2 minutes. The distance is unknown. To convert minutes to seconds, multiply 2 minutes by 60 seconds per minute, yielding 120 seconds. R (floors/sec)× T (sec)= D (floors) Elevator 0.25 × 120 = ? you can solve for the distance using the RT = D equation: 0.25(120) = d d = 30 floors A train travels 90 kilometers/hour. How many hours does it take the train to travel 450,000 meters? (1 kilometer = 1,000 meters) TRY OUT.
Identity 1: ca + cb = c(a + b) Identity 2: ca − cb = c(a − b) Identity 3: (a+b)^2 = a^2 + 2ab+ b^2 Identity 4: (a-b)^2 = a^2 - 2ab+ b^2 Identity 5: a^2 - b^2 = (a+b) (a-b) Identity 6: (a+b)^3 = a^3 + 3a^2b + 3ab^2 Identity 7 (a-b)^3 = a^3 - 3a^2b + 3ab^2
Rules of Exponents https://www.rapidtables.com/math/number/exponent.html
Machine A fills soda bottles at a constant rate of 60 bottles every 12 minutes and Machine B fills soda bottles at a constant rate of 120 bottles every 8 minutes. How many bottles can both machines working together at their respective rates fill in 25 minutes? To answer these questions quickly and accurately, it is a good idea to begin by expressing rates in equivalent units: Rate of Machine A= 60 bottles/12mins= 5 bottles per min; Rate of Machine B= 120 bottles/8mins= 15 bottles per min That means that working together they fill 5 + 15 = 20 bottles every minute. Now you can fill out an RTW chart. Let b be the number of bottles filled: R (bottles/min) × T (min) = W (bottles) Now solve for the b: b = 20 × 25 = 500 bottles
Alejandro, working alone, can build a doghouse in 4 hours. Betty can build the same doghouse in 3 hours. If Betty and Carmelo, working together, can build the doghouse twice as fast as Alejandro, how long would it take Carmelo, working alone, to build the doghouse? The problem states that Betty and Carmelo, working together, can work twice as fast as Alejandro. That means that their rate is twice Alejandro's rate: Rate B + Rate C = 2 ( Rate A) 1/3+ 1/c= 2(1/4) 1/c= ½ - 1/3 = 1/6 It takes Carmelo 6 hours working by himself to build the doghouse
Multiple Rates To deal with this, you will need to deal with multiple RT = D relationships. You can still use the RTD chart. Just add rows. For example: Harvey runs a 30-mile course at a constant rate of 4 miles per hour. If Clyde runs the same track at a constant rate and completes the course in 90 fewer minutes, how fast did Clyde run? To answer these questions correctly, you need to pay attention to the relationships between these two equations. For instance, both Harvey and Clyde ran the same course, so the distance they both ran was 30 miles. Additionally, you know Clyde ran for 90 fewer minutes. To make units match, you can convert 90 minutes to 1.5 hours. If Harvey ran t hours, then Clyde ran (t − 1.5) hours R (miles/hr)× T (hr)= D (miles) Harvey 4 t = 30 so t= 7.5 Clyde rate * t − 1.5= 30 If t = 7.5, then Clyde ran for 7.5 − 1.5 = 6 hours. You can now solve for Clyde's rate. Let r equal Clyde's rate: r * t= 30 R*6=30 R= 5 ans For questions that involve multiple rates, remember to set up multiple RT = D equations and look for relationships between the equations IMPT
Average Rate: Don't Just Add and Divide If Lucia walks to work at a rate of 4 miles per hour, but she walks home by the same route at a rate of 6 miles per hour, what is Lucia's average walking rate for the round trip? To find the average rate, you must first find the total combined time for the trips and the total combined distance for the trips. Set up a Multiple RTD chart: Rate (mi/hr) × Time (hr) = Distance (mi) Going 4 × t1 = 12 Return 6 × t2 = 12 Total r × t3 = 24 The times can be found using the RTD equation. For the Going trip, 4t1 = 12, so t1 is 3 hours. For the Return trip, 6t2 = 12, so t2 is 2 hours. Thus, the total time is 5 hours. Now plug in these numbers: Rate (mi/hr) × Time (hr) = Distance (mi) Going 4 × 3 = 12 Return 6 × 2 = 12 Total r × 5 = 24 Now that you have the total time and the total distance, you can find the average rate using the RTD equation: RT=D R(5)=24 R= 4.8 miles per day ans In fact, it is the weighted average of the two rates, with the times as the weights. Because of that, the average rate is closer to the slower of the two rates.
Solving Linear Inequalities To solve an inequality means to find the set of all values of the variable that make the inequality true. This set of values is also known as the solution set of an inequality. Two inequalities that have the same solution set are called equivalent inequalities. The procedure used to solve a linear inequality is similar to that used to solve a linear equation, which is to simplify the inequality by isolating the variable on one side of the inequality, using the following two rules Rule 1: When the same constant is added to or subtracted from both sides of an inequality, the direction of the inequality is preserved and the new inequality is equivalent to the original. Rule 2: When both sides of the inequality are multiplied or divided by the same nonzero constant, the direction of the inequality is preserved if the constant is positive but the direction is reversed if the constant is negative. In either case, the new inequality is equivalent to the original
Functions An algebraic expression in one variable can be used to define a function of that variable. Functions are usually denoted by letters such as f, g, and h. For example, the algebraic expression 3x + 5 can be used to define a function f by f(x) = 3x + 5 where f(x) is called the value of f at x and is obtained by substituting the value of x in the expression above. For example, if x = 1 is substituted in the expression above, the result is f(1) = 3(1) + 5 = 8.
Graphing Quadratic Equations The graph of a quadratic equation of the form , y= ax^2 + bx +c =0 where a, b, and c are constants and a ≠ 0, is a parabola. The x-intercepts of the parabola are the solutions of the equation ax^2 + bx +c =0 . If a is positive, the parabola opens upward and the vertex is its lowest point. If a is negative, the parabola opens downward and the vertex is its highest point. Every parabola that is the graph of a quadratic equation of the form ax^2 + bx +c =0 is symmetric with itself about the vertical line that passes through its vertex. In particular, the two x-intercepts are equidistant from this line of symmetry
Graphing Circles The graph of an equation of the form (x-a)^2 + (y-b)^2=r^2 is a circle with its center at the point (a, b) and with radius r > 0
linear equations y = mx + b m and b represent numbers (positive, negative, or 0) The numbers m and b have special meanings when you are dealing with linear equations. First, m = slope Slope formula= y2 - y1/x2 -x1 Next, b = y-intercept. This tells you where the line crosses the y-axis. Any line or curve crosses the y-axis when x = 0. To find the y-intercept, plug in 0 for x in the equation:
The Intercepts of a Line A point where a line intersects a coordinate axis is called an intercept. There are two types of intercepts: the x-intercept, where the line intersects the x-axis, and the y-intercept, where the line intersects the y-axis. The x-intercept is expressed using the ordered pair (x, 0), where x is the point where the line intersects the x-axis. The x-intercept is the point on the line at which y = 0. The y-intercept is expressed using the ordered pair (0, y), where y is the point where the line intersects the y-axis. The y-intercept is the point on the line at which x = 0. If two lines in a plane do not intersect, then the lines are parallel
Proportions Simple Ratio problems can be solved with a proportion. For example: The ratio of girls to boys in the class is 4 to 7. If there are 35 boys in the class, how many girls are there? Step 1: Set up a labeled proportion:4 girls/7 boys= xgirls/35 boys Step 2: Cross-multiply to solve: 7x= 3* 35; x= 20
The Unknown Multiplier For more complicated Ratio problems, in which the total of all items is given, the "Unknown Multiplier" technique is useful: The ratio of men to women in a room is 3 : 4. If there are 56 people in the room, how many of the people are men? Men+ women = 56; 3x+4x = 56; 7x=56; x=8 Now you know that the value of x, the Unknown Multiplier, is 8. Therefore, you can determine the exact number of men and women in the room: The number of men equals: 3x = 3(8) = 24. The number of women equals: 4x = 4(8) = 32. IMPT When should you use the Unknown Multiplier? You should use it when - (1) the total items is given, or (2) neither quantity in the ratio is already equal to a number or a variable expression.
Basic Work Problems Work problems are just another type of Rate problem. Instead of distances, however, these questions are concerned with the amount of "work" done. use the equation RT = W. For example, if a machine produces pencils at a constant rate of 120 pencils every 30 seconds, the rate at which the machine works is 120 pencils/30 sec = 4 pencil per sec ans Martha can paint ½ of a room in1 1/4 hours. If Martha finishes painting the room at the same rate, how long will it have taken Martha to paint the room? TRY it
Working Together: Add the Rates When two or more workers are performing the same task, their rates can be added together. For instance, if Machine A can make 5 boxes in an hour, and Machine B can make 12 boxes in an hour, then working together the two machines can make 5 + 12 = 17 boxes per hour. Likewise, if Lucas can complete 1/3 of a task in an hour and Serena can complete 1/2 of that task in an hour, then working together they can complete 1/3 + 1/2 = 5/6 of the task every hour. If, on the other hand, one worker is undoing the work of the other, subtract the rates. For instance, if one hose is filling a pool at a rate of 3 gallons per minute, and another hose is draining the pool at a rate of 1 gallon per minute, the pool is being filled at a rate of 3 − 1 = 2 gallons per minute.
Simple interest Simple interest is based only on the initial deposit, which serves as the amount on which interest is computed, called the principal, for the entire time period. If the amount P is invested at a simple annual interest rate of r percent, then the value V of the investment at the end of t years is given by the formula V= P( 1 + rt/100)
compound interest In the case of compound interest, interest is added to the principal at regular time intervals, such as annually, quarterly, and monthly V= P( 1 + r/100)t If the amount P is invested at an annual interest rate of r percent, compounded n times per year, then the value V of the investment at the end of t years is given by the formula V= P( 1 + r/100n)nt
