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Solving Equations with Variables on Both Sides 7-3 Learn to solve equations with variables on both sides of the equal sign.

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Solving Equations with Variables on Both Sides 7-3 Some problems produce equations that have variables on both sides of the equal sign. Solving an equation with variables on both sides is similar to solving an equation with a variable on only one side. You can add or subtract a term containing a variable on both sides of an equation.

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Solving Equations with Variables on Both Sides 7-3 Solve. 4x + 6 = x Additional Example 1A: Solving Equations with Variables on Both Sides 4x + 6 = x – 4x 6 = –3x Subtract 4x from both sides. Divide both sides by –3. –2 = x 6 –3 –3x –3 =

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Solving Equations with Variables on Both Sides 7-3 Check your solution by substituting the value back into the original equation. For example, 4(2) + 6 = 2 or 2 = 2. Helpful Hint

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Solving Equations with Variables on Both Sides 7-3 Solve. 9b – 6 = 5b + 18 Additional Example 1B: Solving Equations with Variables on Both Sides 9b – 6 = 5b + 18 – 5b 4b – 6 = 18 4b4b = Subtract 5b from both sides. Divide both sides by 4. b = b = 24 Add 6 to both sides.

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Solving Equations with Variables on Both Sides 7-3 Solve. 9w + 3 = 9w + 7 Additional Example 1C: Solving Equations with Variables on Both Sides 3 ≠ 7 9w + 3 = 9w + 7 – 9w Subtract 9w from both sides. No solution. There is no number that can be substituted for the variable w to make the equation true.

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Solving Equations with Variables on Both Sides 7-3 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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Solving Equations with Variables on Both Sides 7-3 Solve. 5x + 8 = x Check It Out: Example 1A 5x + 8 = x – 5x 8 = –4x Subtract 5x from both sides. Divide both sides by –4. –2 = x 8 –4 –4x –4 =

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Solving Equations with Variables on Both Sides 7-3 Solve. 3b – 2 = 2b + 12 – 2b b – 2 = 12 Subtract 2b from both sides. + 2 b = 14 Add 2 to both sides. Check It Out: Example 1B

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Solving Equations with Variables on Both Sides 7-3 Solve. 3w + 1 = 3w ≠ 8 3w + 1 = 3w + 8 – 3w 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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Solving Equations with Variables on Both Sides 7-3 To solve multi-step equations with variables on both sides, first combine like terms and clear fractions. Then add or subtract variable terms to both sides so that the variable occurs on only one side of the equation. Then use properties of equality to isolate the variable.

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Solving Equations with Variables on Both Sides 7-3 Solve. 10z – 15 – 4z = 8 – 2z - 15 Additional Example 2: Solving Multi-Step Equations with Variables on Both Sides 10z – 15 – 4z = 8 – 2z – z – 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 =

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Solving Equations with Variables on Both Sides 7-3 Solve. 12z – 12 – 4z = 6 – 2z + 32 Check It Out: Example 2 12z – 12 – 4z = 6 – 2z z – 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 z =

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Solving Equations with Variables on Both Sides 7-3 Additional Example 3: Business Application Daisy’s Flowers sell a rose bouquet for $39.95 plus $2.95 for every rose. A competing florist sells a similar bouquet for $26.00 plus $4.50 for every rose. Find the number of roses that would make both florists’ bouquets cost the same price.

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Solving Equations with Variables on Both Sides 7-3 Additional Example 3 Continued r = r Let r represent the price of one rose. – 2.95r = r Subtract 2.95r from both sides. – Subtract from both sides = 1.55r r 1.55 = Divide both sides by = r The two services would cost the same when purchasing 9 roses.

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Solving Equations with Variables on Both Sides 7-3 Additional Example 5: Solving Literal Equations for a Variable The equation t = m + 10e gives the test score t for a student who answers m multiple-choice questions and e essay questions correctly. Solve this equation for e. t = m + 10e Locate e in the equation. Since m is added to 10e, subtract m from both sides. t = m + 10e –m –m t – m = 10e 10 t – m = e 10 Since e is multiplied 10, divide both sides by 10.

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Solving Equations with Variables on Both Sides 7-3 Lesson Quiz Solve. 1. 4x + 16 = 2x 2. 8x – 3 = x 3. 2(3x + 11) = 6x x = x – 9 5. An apple has about 30 calories more than an orange. Five oranges have about as many calories as 3 apples. How many calories are in each? x = 6 x = –8 no solution x = An orange has 45 calories. An apple has 75 calories.

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