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Review 3.1-3.4 Solving Equations 3.1 Addition and Subtraction 3.2 Multiplication and Division 3.3 Multi-step equations 3.4 Variables on Both Sides

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1) Solve r + 16 = -7 Think of this equation as a balance scale. Whatever you do to one side has to be done to the other to keep it balanced!

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- 16 -16 r = -23 -23 + 16 = -7 1) Solve r + 16 = -7 1.Subtract 16 from both sides 2.Simplify vertically 3.Check your answer by substituting your answer back into the problem

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2) Solve x + 2 = -3 Get the variable by itself. What is your first step? 1.Add 2 to both sides 2.Subtract 2 from both sides 3.Add 3 to both sides 4.Subtract 3 from both sides Answer Now

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2) Solve x + 2 = -3 - 2 - 2 x = -5 -5 + 2 = -3 1.Subtract 2 from both sides 2.Simplify vertically 3.Check your answer by substituting your answer back into the problem On homework and tests, be sure to check your work!! There is no reason why you should miss a problem!

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3) Solve 8 = m - 3 1.m = 5 2.m = 11 3.m = 24 4.m = 8/3 Answer Now

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3) Solve 8 = m - 3 + 3+ 3 11= m m = 11 8 = 11 - 3 1.Add 3 to both sides 2.Simplify vertically 3.Check your answer by substituting your answer back into the problem

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When solving equations, we want to eliminate double signs. y + (-3) = 8 is rewritten as y – 3 = 8 p – (-5) = 6 is rewritten as p + 5 = 6 As a general rule, replace “+ (- )” with “–” and “– (- )” with “+”. This will make things less confusing in the future!

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4) Solve y + (-3) = 7 y – 3 = 7 + 3 +3 y = 10 10 + (-3) = 7 1.Eliminate the double sign 2.Add 3 to both sides 3.Simplify vertically 4.Check your answer by substituting your answer back into the problem

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5) Solve. -x - (-2) = 1 -x + 2 = 1 - 2 - 2 -x = -1 -1 -1 x = 1 -(1) + 2 = 1 1.Eliminate the double sign 2.Subtract 2 from both sides 3.Simplify vertically 4.We haven’t gotten x by itself. It still has a negative sign in front. 5.Divide bot sides by -1 6.Check your answer

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Solve -y – (-3) = 7 1.y = 10 2.y = 4 3.y = -10 4.y = -4 Answer Now

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-y - (-3) = 7 -y + 3 = 7 -3 -3 -y = 4 y = -4

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1) Solve -5t = 60 -5 -5 t = -12 -5(-12) = 60 1.Divide both sides by -5 2.Simplify 3.Check your answer

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2) Solve 15 = 6n 6 6 5/2 = n n = 5/2 15 = 6(5/2) 1.Divide both sides by 6 2.Simplify 3.Check your answer

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3) Solve -3v = -129 1.v = -126 2.v = -43 3.v = 43 4.v = 126 Answer Now

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4) Solve 4 · · 4 x = -48 1.Clear the fraction – multiply both sides by 4 2.Simplify 3.Check your answer

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(3/2) · = 18 · (3/2) x = 54/2 x = 27 1.Clear the fraction – multiply both sides by the RECIPRICAL 2.Simplify 3.Check your answer 5) Solve = 18

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6) Which step clears the fraction in 1.Multiply by 3/5 2.Multiply by 5/3 3.Multiply by -12 4.Multiply by -5 Answer Now

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7) Solve 1.b = -56 2.b = -14 3.b = 14 4.b = 56 Answer Now

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1) Solve 2x - 1 = -3 + 1 + 1 2x = -2 2 2 x = -1 2(-1) - 1 = -3 -2 – 1 = -3 1.Add 1 to both sides 2.Simplify 3.Divide both sides by 2 4.Simplify 5.Check your answer

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+ 4 + 4 3 · · 3 x = 36 12 – 4 = 8 2) Solve 1.Add 4 to both sides 2.Simplify 3.Clear the fraction - Multiply both sides by 3 4.Simplify 5.Check your answer

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3) Solve 3y – 1 = 8 1.y = 3 2.y = -3 3.y = 4.y = Answer Now

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3) Solve d – 4 = 6 + 4 + 4 d = 10 1.Clear the fraction - Multiply both sides by 2 2.Simplify 3.Add 4 to both sides 4.Simplify 5.Check your answer

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4) Solve 1.d = -7 2.d = -19 3.d = -17 4.d = 17 Answer Now

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5) Solve -3 -3 a = 35 1.Subtract 3 from both sides 2.Simplify 3.Clear the fraction – Multiply both sides by -7 4.Simplify 5.Check your answer

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6) Solve 5z + 16 = 51 1.z = -35 2.z = -7 3.z = 35 4.z = 7 Answer Now

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- 1 - 1 5 · · 5 3x = 15 3 3 x = 5 7) Solve 1.Subtract 1 from both sides 2.Simplify 3.Clear the fraction - Multiply both sides by 5 4.Simplify 5.Divide both sides by 3 6.Simplify 7.Check your answer

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Key Skills Keep it balanced! Solve 6n + 17 = 5 + 4n – 4n 2n + 17 = 5 – 17 2n = -12 ÷ 2 Subtract 4n from each side of the equal sign. Subtract 17 from each side of the equal sign. Divide by 2 n = – 6

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Do This Together Keep it balanced! Solve 3x + 19 = 4 + 8x x = 3 Subtract 3x from each side of the equal sign. – 3x 19 = 4 + 5x Subtract 4 from each side of the equal sign. – 4 15 = 5x Divide by 5 ÷ 5

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Try This Keep it balanced! Solve – 3x – 9 = – 13 – 7x x = – 1 Add 3x to each side of the equal sign. + 7x 4x – 9 = – 13 Add 13 to each side of the equal sign. + 9 4x = – 4 Divide by – 4 ÷ 4

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Do this together Solve 4b + 3 = 3b + 6 3 = b Subtract 4b from each side of the equal sign. – 4b 3 = – b + 6 Subtract 6 from each side of the equal sign. – 6 – 3 = – b Divide by – 1 ÷ – 1

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Two special cases: 6(4 + y) - 3 = 4(y - 3) + 2y 24 + 6y - 3 = 4y - 12 + 2y 21 + 6y = 6y - 12 - 6y - 6y 21 = -12 Never true! 21 ≠ -12 NO SOLUTION! 3(a + 1) - 5 = 3a - 2 3a + 3 - 5 = 3a - 2 3a - 2 = 3a - 2 -3a -2 = -2 Always true! We write IDENTITY.

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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

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

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Solve: Distribute first. 5 + 2y + 8 = 5y - 15 + 10 Next, combine like terms.2y + 13 = 5y – 5 Now solve. (Subtract 2y.)13 = 3y - 5 (Add 5.)18 = 3y (Divide by 3.)6 = y y = 6 5 + 2(y + 4) = 5(y – 3) + 10

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4) 3 - 2x = 4x - 6 3 = 6x - 6 9 = 6x 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.

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Try a few on your own: 9x + 7 = 3x - 5 8 - 2(y + 1) = -3y + 1 8 - 1 z = 1 z - 7 2 4 x = -2 y = -5 z = 20

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1) Solve 3x + 2 = 4x - 1 - 3x 2 = x - 1 + 1 + 1 3 = x 3(3) + 2 = 4(3) - 1 9 + 2 = 12 - 1 1.Subtract 3x from both sides 2.Simplify 3.Add 1 to both sides 4.Simplify 5.Check your answer

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2) Solve 8y - 9 = -3y + 2 + 3y + 3y 11y – 9 = 2 + 9 + 9 11y = 11 11 11 y = 1 8(1) - 9 = -3(1) + 2 1.Add 3y to both sides 2.Simplify 3.Add 9 to both sides 4.Simplify 5.Divide both sides by 11 6.Simplify 7.Check your answer

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1.-3 2. 3. 4.3 What is the value of x if 3 - 4x = 18 + x? Answer Now

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3) Solve 4 = 7x - 3x 4 = 4x 4 4 1 = x 4 = 7(1) - 3(1) 2.Combine like terms 3.Divide both sides by 4 4.Simplify 5.Check your answer

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4) Solve -7(x - 3) = -7 -7x + 21 = -7 - 21 - 21 -7x = -28 -7 -7 x = 4 -7(4 - 3) = -7 -7(1) = -7 1.Distribute 2.Subtract 21 from both sides 3.Simplify 4.Divide both sides by -7 5.Simplify 6.Check your answer

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What is the value of x if 3(x + 4) = 2(x - 1)? Answer Now 1.-14 2.-13 3.13 4.14

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3 - 2x = 4x – 6 + 2x +2x 3 = 6x – 6 + 6 + 6 9 = 6x 6 6 or 1.5 = x 5) Solve 1.Clear the fraction – multiply each term by the LCD 2.Simplify 3.Add 2x to both sides 4.Simplify 5.Add 6 to both sides 6.Simplify 7.Divide both sides by 6 8.Simplify 9.Check your answer

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Special Case #1 6) 2x + 5 = 2x - 3 -2x 5 = -3 This is never true! No solutions 1.Subtract 2x from both sides 2.Simplify

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Special Case #2 7) 3(x + 1) - 5 = 3x - 2 3x + 3 – 5 = 3x - 2 3x - 2 = 3x – 2 -3x -3x -2 = -2 This is always true! Infinite solutions or identity 1.Distribute 2.Combine like terms 3.Subtract 3x from both sides 4.Simplify

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What is the value of x if -3 + 12x = 12x - 3? 1.0 2.4 3.No solutions 4.Infinite solutions Answer Now

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Challenge! What is the value of x if -8(x + 1) + 3(x - 2) = -3x + 2? Answer Now 1.-8 2.-2 3.2 4.8

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