CLASS NOTES: 2 1 thru 2 5 Solving Equations

Transcription

1 Page 1 of 13 CLASS NOTES: 2 1 thru 2 5 Solving Equations 2 1: Solving One-Step Equations Vocabulary: Equivalent equations equations that have the same solution Isolate get by itself Inverse operation operations that undo each other. Add and subtract are inverse operations. Multiply and divide are inverse operations. Addition Property of Equality Adding the same number to each side of an equation produces an equivalent equation. Algebra Example If, then Subtraction Property of Equality Subtracting the same number from each side of an equation produces an equivalent equation. Algebra Example If, then EX 1 Use the subtraction property of equality to solve each equation. (a) x + 13 = 27 (b) y + 2 = 6

2 Page 2 of 13 EX 2 Use the addition property of equality to solve each equation. (a) 7 = b 3 (b) m 8 = 14 (c) 1 2 = y 3 2 Multiplication Property of Equality Multiplying each side of an equation by the same number produces an equivalent equation. Algebra Example If, then Division Property of Equality Dividing each side of an equation by the same number produces an equivalent equation. Algebra Example If, then

3 Page 3 of 13 EX 3 Use the division property of equality to solve each equation. (a) 4x = 6.4 (b) 10 = 15x (c) 3.2z = 14 EX 4 Use the multiplication property of equality to solve each equation. (a) x 4 = 9 (b) 19 = r 3 (c) x 9 = 8 EX 5 Use reciprocals to solve each equation. (a) 4 5 m = 28 (b) 12 = 3 4 x

4 Page 4 of : Solving Two-Step Equations Steps for Solving a Two-Step Equation First, add or subtract on both sides of the equation. Second, multiply or divide on both sides of the equation. EX 1 Solve each equation using two steps. (a) 2x + 3 = 15 (b) 5x + 12 = 13 (c) 5 = t 2 3 (d) 6 = m 7 3 (c) x 7 3 = 12 What operation should you perform first? Multiplication. When you multiply by the denominator of the fraction in the equation, you get a new onestep equation with no fraction.

5 Page 5 of 13 EX 2 Solve each equation using two steps. (a) 6 = x 2 4 EX 3 Solve each equation using two steps. State the rule that justifies each step. (a) t + 8 = 3 (b) x 3 5 = 4

6 Page 6 of 13 EX 4 (a) You are making a bulletin board to advertise community service opportunities for students. You plan to use half a sheet of construction paper for each ad. You need five sheets of construction paper for a title banner. You have 18 sheets of construction paper. How many ads can you make? (b) Suppose you used one quarter of a sheet of paper for each ad and four full sheets for the title banner in the last problem. How many ads could you make.

7 Page 7 of : Solving Multi-Step Equations EX 1 Solve each multi-step equation. (a) 5 = 5m m How is this equation different from equations you ve seen before? The variable occurs in two terms. You can simplify the equation by grouping like terms and combining them. (b) 11m 8 6m = 22 (c) 2y y = 14 EX 2 Noah and Kate are shopping for new guitar strings in a music store. Noah buys 2 packs of strings. Kate buys 2 packs of strings and a music book. The book costs $16. Their total cost is $72. How much is one pack of strings?

8 Page 8 of 13 EX 3 Solve each multi-step equation. (a) 8 ( 2x 1) = 36 How can you make the equation easier to solve? Remove the grouping symbol by using the Distributive Property. (b) 18 = 3( 2x 6) EX 4 Solve each multi-step equation with fractions. (a) 3x 4 x 3 = 10 How do you get started? Clear the fractions from the equation by multiplying by the number that is the common denominator of all the fractions in the equation. Multiply both sides by 12 Distribute the 12 on the left Now multiply Combine like terms Divide

9 Page 9 of 13 EX 5 Solve each multi-step equation with fractions. (a) 2b 5 + 3b 4 = 3 (b) 1 9 = 5 6 m 3 EX 6 Solve the equation that contains decimals. (a) x = 1.24 EX 7 Martha takes her niece and nephew to a concert. She buys T-shirts and bumper stickers for them. The bumper stickers cost $1 each. Martha s niece wants 1 shirt and 4 bumper stickers, and her nephew wants 2 shirts but no bumper stickers. If Martha s total is $67, what is the cost of one shirt?

10 Page 10 of : Solving Equations With Variables on Both Sides EX 1 Solve equations with variables on both sides. (a) 5x + 2 = 2x + 14 How do you get started? Add or subtract to get all the variables on one side only. (b) 7k + 2 = 4k 10 (c) 2( 5x 1) = 3( x + 11) How do you get started? First, remove the parentheses by using the distributive property. Then add or subtract to move all variables to one side. (d) 4 ( 2y + 1) = 2( y 13)

11 Page 11 of 13 EX 2 What is the solution of each equation? (a) 10x + 12 = 2( 5x + 6) How can you tell how many solutions an equation has? If you eliminate the variable in the process of solving, the equation is either an identity with infinitely many solutions or an equation with no solution. Identity: an equation that is exactly the same on both sides. (b) 9m 4 = 3m m (c) 3( 4b 2) = b (d) 2x + 7 = 1( 3 2x ) Steps for Solving Equations 1. Multiply by the common denominator to remove fractions. Use the Distributive Property to remove parentheses. 2. Combine like terms on each side. 3. Add or subtract to get all of the variables on one side. 4. Add, subtract, multiply or divide on both sides to solve for the variable. 5. Check your answer by plugging it back in to the original equation.

12 Page 12 of : Literal Equations and Formulas Vocabulary: Literal equation an equation that has two or more variables. Formula an equation that states a relationship among quantities. EX 1 Rewrite the literal equation. (a) 10x + 5y = 80; for y What is y when x = 3,6? (b) 4 = 2m 5n; for m What is m when n = 2,0,2?

13 Page 13 of 13 EX 2 Rewrite the literal equation. (a) ax + by = c ; for x (b) t = r + px ; for x EX 3 Rewrite the formula. (a) What is the radius of a circle with circumference 64 ft? Round to the nearest tenth. Use 3.14 for π. (b) What is the height of a triangle that has an area of 24 in 2 and a base with a length of 8 in? (c) The monarch butterfly is the only butterfly that migrates annually north and south. The distance that a particular group of monarch butterflies travels is 1,700 miles one way. It takes a typical butterfly about 120 days to travel one way. What is the average rate at which a butterfly travels in miles per day? Round to the nearest mile per day.