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

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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. Course Solving Equations with Variables on Both Sides

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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 = Course Solving Equations with Variables on Both Sides

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Course Solving Equations with Variables on Both Sides 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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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. Course Solving Equations with Variables on Both Sides

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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. Course Solving Equations with Variables on Both Sides

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Course 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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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 = Course Solving Equations with Variables on Both Sides

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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 Course Solving Equations with Variables on Both Sides

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Solve. 3w + 1 = 3w w + 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 Course Solving Equations with Variables on Both Sides

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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. Course Solving Equations with Variables on Both Sides

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Solve. 10z – 15 – 4z = 8 – 2z - 15 Additional Example 2A: 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 = Course Solving Equations with Variables on Both Sides

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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. y5y y53y – = y – y5y y53y – = y – 20 ( ) = 20 ( ) y5y y53y – y – 20 ( ) + 20 ( ) – 20 ( ) = 20(y) – 20 ( ) y5y5 3y53y Course Solving Equations with Variables on Both Sides

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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. Course Solving Equations with Variables on Both Sides

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

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Multiply by the LCD, 24. 6y + 20y + 18 = 24y – 18 26y + 18 = 24y – 18Combine like terms. y4y y65y = y – y4y y65y ( ) = 24 ( ) y4y y65y y – 24 ( ) + 24 ( ) + 24 ( ) = 24(y) – 24 ( ) y4y4 5y65y Check It Out: Example 2B Course Solving Equations with Variables on Both Sides

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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 Course Solving Equations with Variables on Both Sides

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Additional Example 4 Continued Now find the amount of money Jamie spends each morning d Choose one of the original expressions. Jamie spends $1.75 each morning (0.25) = n = Let n represent the number of doughnuts. Find the number of doughnuts Jamie buys on Tuesday. 0.25n = 1.75 n = 7; Jamie bought 7 doughnuts on Tuesday. Divide both sides by Course Solving Equations with Variables on Both Sides

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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 Insert Lesson Title Here no solution x = An orange has 45 calories. An apple has 75 calories. Course Solving Equations with Variables on Both Sides

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