Homework Answers: Workbook page 2 N, W, Z, Q, R 13) Assoc. (X) 25) 8x – y – 3 I,R 14) Comm. (X) 26) -4c I, R 15) Add. Inverse 27) -5r – 58s W, Z, Q, R 16) Mult. Inverse 28) 4a + 1 Q, R 17) Distributive 29) -12 – 8x + 6y Z, Q, R 18) Add. Identity 30) 13y Z, Q, R 19) -0.4, 2.5 or 5/2 31) (110t + 100) mi Q, R 20) 1.6, -0.625 or -5/8 32) false; counter- Comm. (+) 21) 11/16, -16/11 example: Assoc. (+) 22) -5 5/6, 6/35 5(1/5) is not > Mult. Identity 23) 3x 4(1/4) Distributive 24) -4a-16b

Solving Equations Worksheet K = -2 7) x = 19 H = -4 8) x = 7 D = -16/13 or -1.231 9) x = -1/15 or .0667 J = -63/2 or -31.5 10) k = 18/23 or.7826 X = 27 11) r = -5 X = 36 12) g = -16

Solving Equations with the Variable on Both Sides Objectives: to solve equations with the variable on both sides. to solve equations containing grouping symbols. A.1 Solve linear equations in one variable. A.1 Apply these skills to solve practical problems. A.3 Justify steps used in solving equations.

To solve equations with variables on both sides: 1. Use the addition property to move all variables to one side of the equal sign. 2. Solve the equation by working the problem backwards.

Let’s see a few examples: 1) 6x - 3 = 2x + 13 -2x -2x 4x - 3 = 13 +3 +3 4x = 16 4 4 x = 4 Be sure to check your answer! 6(4) - 3 =? 2(4) + 13 24 - 3 =? 8 + 13 21 = 21

Let’s try another! Check: 2) 3n + 1 = 7n - 5 -3n -3n 3(1.5) + 1 =? 7(1.5) - 5 4.5 + 1 =? 10.5 - 5 5.5 = 5.5 2) 3n + 1 = 7n - 5 -3n -3n 1 = 4n - 5 +5 +5 6 = 4n 4 4 Reduce! 3 = n 2

Here’s a tricky one! 3) 5 + 2(y + 4) = 5(y - 3) + 10 Distribute first. Next, combine like terms. 2y + 13 = 5y - 5 Now solve. (Subtract 2y.) 13 = 3y - 5 (Add 5.) 18 = 3y (Divide by 3.) 6 = y Check: 5 + 2(6 + 4) =? 5(6 - 3) + 10 5 + 2(10) =? 5(3) + 10 5 + 20 =? 15 + 10 25 = 25

Let’s try one with fractions! 3 - 2x = 4x - 6 3 = 6x - 6 9 = 6x so x = 3/2 Steps: Multiply each term by the least common denominator (8) to eliminate fractions. Solve for x. Add 2x. Add 6. Divide by 6.

Two special cases: 6(4 + y) - 3 = 4(y - 3) + 2y 21 = -12 Never true! 21 ≠ -12 NO SOLUTION! 3(a + 1) - 5 = 3a - 2 3a + 3 - 5 = 3a - 2 3a - 2 = 3a - 2 -3a -3a -2 = -2 Always true! We write ALL REAL NUMBERS.

Try a few on your own: 9x + 7 = 3x - 5 8 - 2(y + 1) = -3y + 1 8 - 1 z = 1 z - 7 2 4

The answers: x = -2 y = -5 z = 20

Solving Formulas: What it means to solve To solve for x would mean to get x by itself on one side of the equation, with no x’s on the other side. (x = __ ) Similarly, to solve for y would mean to get y by itself on one side of the equation, with no y’s on the other side. (y = __ )

The DO-UNDO chart 1) Solve the equation -5x + y = -56 for x. Ask yourself: What is the first thing being done to x, the variable being solved for? x is being multiplied by -5. DO UNDO ·-5 - y What is being done next? + y ÷(-5) y is being added to -5x.

Show all of your work! First, subtract y from both sides of the equation. Next, divide by -5. This process actually requires LESS WORK than solving equations in one variable  Ex: -5x + y = -56 - y -y -5x = -56 - y -5 -5 x = -56 - y = 56 + y -5 5

Let’s try another: Ex: Solve 2x - 4y = 7 for x. +4y + 4y 2x = 7 + 4y 2 2 x = 7 + 4y 2 This fraction cannot be simplified unless both terms in the numerator are divisible by 2. Complete the do-undo chart. DO UNDO · 2 + 4y - 4y ÷ 2 To solve for x: First add 4y Then divide by 2

Another example: Solve a(y + 1) = b for y. DO UNDO + 1 ÷ a · a - 1 To solve for y: First divide by a Then subtract 1 a(y + 1) = b a a y + 1 = b a - 1 -1 y = b - 1

Here’s a tricky one! Solve 3ax - b = d - 4cx for x. First, we must get all terms with x together on one side. Add 4cx to both sides Add b to both sides Next, use the distributive property to factor x out of the two terms on the left. Now, x is being multiplied by (3a + 4c). To undo this, divide both sides by (3a + 4c). 3ax - b = d - 4cx +4cx +4cx 3ax - b + 4cx = d +b +b 3ax + 4cx = d + b x(3a + 4c) = d + b (3a + 4c) (3a + 4c) x = d + b (3a + 4c)

Try a few on your own. Solve P = 1.2W for W. H2 Solve P = 2l + 2w for l. Solve 4x - 3m = 2mx - 5 for x.

The answers: DO UNDO · 1.2 · H2 ÷ H2 ÷ 1.2 W = PH2 1.2 DO UNDO · 2 -2w l = P - 2w 2 Subtract 2mx, then Add 3m to get 4x - 2mx = 3m - 5 x(4 - 2m) = 3m - 5 Divide by (4 - 2m) x = 3m - 5 4 - 2m