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3-4 Solving Equations with Variables on Both Sides Extension of AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. Also covered: AF1.1 California Standards

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3-4 Solving Equations with Variables on Both Sides Solve. 4x + 6 = x Additional Example 1A: Solving Equations with Variables on Both Sides 4x + 6 = x – 4x 6 = –3x To collect the variable terms on one side, subtract 4x from both sides. Since x is multiplied by -3, divide both sides by –3. –2 = x 6 –3 –3x –3 =

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3-4 Solving Equations with Variables on Both Sides You can always check your solution by substituting the value back into the original equation. Helpful Hint

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3-4 Solving Equations with Variables on Both Sides Solve. 9b – 6 = 5b + 18 Additional Example 1B: Solving Equations with Variables on Both Sides 9b – 6 = 5b + 18 – 5b 4b – 6 = 18 4b4b 4 24 4 = To collect the variable terms on one side, subtract 5b from both sides. Since b is multiplied by 4, divide both sides by 4. b = 6 + 6 4b = 24 Since 6 is subtracted from 4b, add 6 to both sides.

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3-4 Solving Equations with Variables on Both Sides Solve. 9w + 3 = 9w + 7 Additional Example 1C: Solving Equations with Variables on Both Sides 3 7 9w + 3 = 9w + 7 – 9w To collect the variable terms on one side, subtract 9w from both sides. There is no solution. There is no number that can be substituted for the variable w to make the equation true.

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3-4 Solving Equations with Variables on Both Sides if the variables in an equation are eliminated and the resulting statement is false, the equation has no solution. Helpful Hint

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3-4 Solving Equations with Variables on Both Sides Solve. 5x + 8 = x Check It Out! Example 1A 5x + 8 = x – 5x 8 = –4x Since x is multiplied by –4, divide both sides by –4. –2 = x 8 –4 –4x –4 = To collect the variable terms on one side, subtract 5x from both sides.

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3-4 Solving Equations with Variables on Both Sides Solve. 3b – 2 = 2b + 12 – 2b b – 2 = 12 + 2 b = 14 Since 2 is subtracted from b, add 2 to both sides. Check It Out! Example 1B To collect the variable terms on one side, subtract 2b from both sides.

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3-4 Solving Equations with Variables on Both Sides Solve. 3w + 1 = 3w + 8 1 8 3w + 1 = 3w + 8 – 3w To collect the variable terms on one side, subtract 3w from both sides. No solution. There is no number that can be substituted for the variable w to make the equation true. Check It Out! Example 1C

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3-4 Solving Equations with Variables on Both Sides To solve more complicated equations, you may need to first simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.

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3-4 Solving Equations with Variables on Both Sides Solve. 10z – 15 – 4z = 8 – 2z – 15 Additional Example 2A: Solving Multi-Step Equations with Variables on Both Sides 10z – 15 – 4z = 8 – 2z – 15 + 15 +15 6z – 15 = –2z – 7Combine like terms. + 2z Add 2z to both sides. 8z – 15 = – 7 8z = 8 z = 1 Add 15 to both sides. Divide both sides by 8. 8z 8 8 8 =

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3-4 Solving Equations with Variables on Both Sides Additional Example 2B: Solving Multi-Step Equations with Variables on Both Sides Multiply by the LCD, 20. 4y + 12y – 15 = 20y – 14 16y – 15 = 20y – 14Combine like terms. y5y5 3434 3y53y5 7 10 + – = y – y5y5 3434 3y53y5 7 10 + – = y – 20 ( ) = 20 ( ) y5y5 3434 3y53y5 7 10 + – y – 20 ( ) + 20 ( ) – 20 ( ) = 20(y) – 20 ( ) y5y5 3y53y5 3434 7 10

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3-4 Solving Equations with Variables on Both Sides Additional Example 2B Continued Add 14 to both sides. –15 = 4y – 14 –1 = 4y + 14 –1 4 4y4y 4 = Divide both sides by 4. –1 4 = y 16y – 15 = 20y – 14 – 16y Subtract 16y from both sides.

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3-4 Solving Equations with Variables on Both Sides Solve. 12z – 12 – 4z = 6 – 2z + 32 Check It Out! Example 2A 12z – 12 – 4z = 6 – 2z + 32 + 12 +12 8z – 12 = –2z + 38Combine like terms. + 2z Add 2z to both sides. 10z – 12 = 38 10z = 50 z = 5 Add 12 to both sides. Divide both sides by 10. 10z 50 10 =

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3-4 Solving Equations with Variables on Both Sides Multiply by the LCD, 24. 6y + 20y + 18 = 24y – 18 26y + 18 = 24y – 18Combine like terms. y4y4 3434 5y65y6 6868 + + = y – y4y4 3434 5y65y6 6868 24 ( ) = 24 ( ) y4y4 3434 5y65y6 6868 + + y – 24 ( ) + 24 ( ) + 24 ( ) = 24(y) – 24 ( ) y4y4 5y65y6 3434 6868 Check It Out! Example 2B

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3-4 Solving Equations with Variables on Both Sides Subtract 18 from both sides. 2y + 18 = – 18 2y = –36 – 18 –36 2 2y2y 2 = Divide both sides by 2. y = –18 26y + 18 = 24y – 18 – 24y Subtract 24y from both sides. Check It Out! Example 2B Continued

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3-4 Solving Equations with Variables on Both Sides Check It Out! Example 3 Marlas Gift Baskets sells a muffin basket for $22.00 plus $2.25 for every balloon. A competing service sells a similar muffin basket for $16.00 plus $3.00 for every balloon. Find the number of balloons that would make both baskets cost the same price. Marlas: c = 22.00 + 2.25 b total cost is flat fee plus cost for each balloon Other: c = 16.00 + 3.00 b Write an equation for each service. Let c represent the total cost and b represent the number of balloons.

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3-4 Solving Equations with Variables on Both Sides Check It Out! Example 3 Continued 22.00 + 2.25b = 16.00 + 3.00b Now write an equation showing that the costs are equal. – 2.25b 22.00 = 16.00 + 0.75b Subtract 2.25b from both sides. – 16.00 Subtract 16.00 from both sides. 6.00 = 0.75b 6.00 0.75 0.75b 0.75 = Divide both sides by 0.75. 8 = b The two services would cost the same when purchasing a muffin basket with 8 balloons.

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