CCGPS Coordinate Algebra EOCT Review Units 1 and 2
Unit 1: Relationships Among Quantities Key Ideas
Unit Conversions A quantity is a an exact amount or measurement. A quantity can be exact or approximate depending on the level of accuracy required.
Ex 1: Convert 5 miles to feet. 5miles 5280feet 1mile 26,400feet
Ex: 2 Convert 50 pounds to grams 50 lbs. 454 grams 1 1 lb. 22,700 grams
Ex: 3 Convert 60 miles per hour to feet per minute. 60miles 1hour 5280 60min feet hr 1mile 5280 ft min
Tip There are situations when the units in an answer tell us if the answer is wrong. For example, if the question called for weight and the answer is given in cubic feet, we know the answer cannot be correct.
4. Review Examples The formula for density d is d = m/v where m is mass and v is volume. If mass is measured in kilograms and volume is measured in cubic meters, what is the unit rate for density? kg m 3
Expressions, Equations & Inequalities Arithmetic expressions are comprised of numbers and operation signs. Algebraic expressions contain one or more variables. The parts of expressions that are separated by addition or subtraction signs are called terms. The numerical factor is called the coefficient.
Example 5: 4x 2 +7xy 3 It has three terms: 4x 2, 7xy, and 3. For 4x 2, the coefficient is 4 and the variable factor is x. For 7xy, the coefficient is 7 and the variable factors are x and y. The third term, 3, has no variables and is called a constant.
Example 6: The Jones family has twice as many tomato plants as pepper plants. If there are 21 plants in their garden, how many plants are pepper plants? How should we approach the solution to this equation? tomato plant: 2x 2x x 21 pepper plant: x x 7
Example 7: Find 2 consecutive integers whose sum is 225. first: x second: x + 1 x x 1 225 x 112 112&113
Example 8: A rectangle is 7 cm longer than it is wide. Its perimeter is at least 58 cm. What are the smallest possible dimensions for the rectangle? 4x 14 58 x 11 11 by 16
Writing Linear & Exponential Equations If the numbers are going up or down by a constant amount, the equation is a linear equation and should be written in the form y = mx + b. If the numbers are going up or down by a common multiplier (doubling, tripling, etc.), the equation is an exponential equation and should be written in the form y = a(b) x.
Create the equation of the line for each of the following tables. 9) 10) x y 0 2 1 6 2 18 3 54 x y 0-5 1 3 2 11 3 19 y 2(3) x y 8x 5
11. Linear Word Problem Enzo is celebrating his birthday and his mom gave him $50 to take his friends out to celebrate. He decided he was going to buy appetizers and desserts for everyone. It cost 5 dollars per dessert and 10 dollars per appetizer. Enzo is wondering what kind of combinations he can buy for his friends. a) Write an equation using 2 variables to represent Enzo s purchasing decision. 5a 10d 50 (Let a = number of appetizers and d = number of desserts.) b) Use your equation to figure out how many desserts Enzo can get if he buys 4 appetizers. 5 4 10d 50 d 3 c) How many appetizers can Enzo buy if he buys 6 desserts? 5a 10 6 50 a 2
12. Exponential Word Problem: Ryan bought a car for $20,000 that depreciates at 12% per year. His car is 6 years old. How much is it worth now? y P 1 r t y 20,000 1.12 6 y $9,288.08
Solving Exponential Equations If the bases are the same, you can just set the exponents equal to each other and solve the resulting linear equation. If the bases are not the same, you must make them the same by changing one or both of the bases. Distribute the exponent to the given exponent. Then, set the exponents equal to each other and solve.
Solve the exponential equation: 13) 4x 8 x 7 14) 2 x x 2 2 2 3 27 4x 8 x 7 x 5 2x 3 x 2 3 3 2x 3 x 2 x 6
Unit 2: Solving Systems of Equations Key Ideas
Reasoning with Equations & Inequalities Understanding how to solve equations Solve equations and inequalities in one variable Solve systems of equations Represent and solve equations and inequalities graphically.
Important Tips Know the properties of operations Be familiar with the properties of equality and inequality. (Watch out for the negative multiplier.) Eliminate denominators (multiply by denominators to eliminate them)
Properties to know Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Commutative Property of Addition and Multiplication Associative Property of Addition and Multiplication Distributive Property Identity Property of Addition and Multiplication Multiplicative Property of Zero Additive and Multiplicative Inverses
Example 15 Solve the equation 8(x + 2) = 2(y + 4) for y. y 4x 4
Example 16 Karla wants to save up for a prom dress. She figures she can save $9 each week from the money she earns babysitting. If she plans to spend up to $150 for the dress, how many weeks will it take her to save enough money? 17weeks
Example 17 This equation can be used to find h, the number of hours it takes Bill and Bob to clean their rooms. h h 5 20 1 4h h 20 h 4 How many hours will it take them?
Example 18 You are selling tickets for a basketball game. Student tickets cost $3 and general admission tickets cost $5. You sell 350 tickets and collect $1450. Use a system of linear equations to determine how many student tickets you sold? Student : x General :y x y 350 3x 5y 1450 150 student
Example 19 You sold 52 boxes of candy for a fundraiser. The large size box sold for $3.50 each and the small size box sold for $1.75 each. If you raised $112.00, how many boxes of each size did you sell? large : x A. 40 large, 12 small B. 12 large, 40 small C. 28 large, 24 small D. 24 large, 28 small small :y x y 52 3.5x 1.75y 112
Example 20 You sold 61 orders of frozen pizza for a fundraiser. The large size sold for $12 each and the small size sold for $9 each. If you raised $660.00, how many of each size did you sell? A. 24 large, 37 small B. 27 large, 34 small C. 34 large, 27 small D. 37 large, 24 small large : x small :y x y 61 12x 9y 660
Example 21 Which equation corresponds to the graph shown? A. y = x + 1 B. y = 2x + 1 C. y = x 2 D. y = -3x 2
Example 22 Which graph would represent a system of linear equations that has no common coordinate pairs? A B C D
Ex. 23 Graph y x x 2 2