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Solve Equations with Variables on Both Sides

Learn how to solve equations that have variables on both sides by collecting variable terms on one side and constant terms on the other side. Includes examples and practice problems.

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Solve Equations with Variables on Both Sides

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  1. Chapter 2.5 Solve Equations with Variables on Both Sides

  2. Key Concept • Some equations have variables on both sides. To solve such equations, you can collect the variable terms on one side of the equation and the constant terms on the other side of the equation.

  3. ANSWER The solution is 2. Check by substituting 2 for xin the original equation. Solve an equation with variables on both sides EXAMPLE 1 Solve 7 – 8x= 4x– 17. 7 – 8x = 4x – 17 Write original equation. 7 – 8x + 8x = 4x – 17 + 8x Add 8xto each side. 7 = 12x – 17 Simplify each side. 24 = 12x Add 17 to each side. 2 = x Divide each side by 12.

  4. ? 7 – 8(2) = 4(2) – 17 ? –9 = 4(2) – 17 –9=–9 EXAMPLE 1 Solve an equation with variables on both sides CHECK 7 – 8x= 4x– 17 Write original equation. Substitute 2 for x. Simplify left side. Simplify right side. Solution checks.

  5. (16x + 60). Solve 9x–5= (16x + 60) 9x – 5= 1 1 4 4 EXAMPLE 2 Solve an equation with grouping symbols Write original equation. 9x – 5 = 4x + 15 Distributive property 5x – 5 = 15 Subtract 4xfrom each side. 5x =20 Add 5 to each side. x = 4 Divide each side by 5.

  6. ANSWER 3 On Your Own GUIDED PRACTICE Solve the equation. Check your solution. 1. 24 – 3m= 5m

  7. ANSWER 9 On Your Own GUIDED PRACTICE Solve the equation. Check your solution. 2. 20 +c= 4c – 7

  8. ANSWER –8 On Your Own GUIDED PRACTICE Solve the equation. Check your solution. 3. 9 – 3k= 17k – 2k

  9. ANSWER 6 On Your Own GUIDED PRACTICE Solve the equation. Check your solution. 4. 5z– 2= 2(3z – 4)

  10. ANSWER 2 On Your Own GUIDED PRACTICE Solve the equation. Check your solution. 5. 3 – 4a= 5(a – 3)

  11. 6. (6y + 15) 8y–6= ANSWER 4 2 3 On Your Own GUIDED PRACTICE Solve the equation. Check your solution.

  12. NUMBER OF SOLUTIONS • Equations do not always have one solution. An equation that is true for all values of the variable is an identity . So, the solution of an identity is all real numbers. Some equations have no solution.

  13. a. 3x = 3(x + 4) b.2x + 10 = 2(x + 5) Identify the number of solutions of an equation EXAMPLE 4 Solve the equation, if possible. SOLUTION a. 3x = 3(x + 4) Original equation 3x = 3x + 12 Distributive property The equation 3x = 3x + 12 is not true because the number 3xcannot be equal to 12 more than itself. So, the equation has no solution. This can be demonstrated by continuing to solve the equation.

  14. 0 = 12 ANSWER The statement 0 = 12 is not true, so the equation has no solution. Identify the number of solutions of an equation EXAMPLE 4 3x – 3x= 3x + 12 – 3x Subtract 3xfrom each side. Simplify.

  15. ANSWER Notice that the statement 2x + 10 = 2x + 10 is true for all values of x. So, the equation is an identity, and the solution is all real numbers. Identify the number of solutions of an equation EXAMPLE 1 EXAMPLE 4 b.2x + 10 = 2(x + 5) Original equation 2x + 10 = 2x + 10 Distributive property

  16. ANSWER no solution On Your Own GUIDED PRACTICE Solve the equation, if possible. 8. 9z + 12= 9(z + 3)

  17. ANSWER 0 On Your Own GUIDED PRACTICE Solve the equation, if possible. 9. 7w + 1= 8w + 1

  18. ANSWER identity On Your Own GUIDED PRACTICE Solve the equation, if possible. 10. 3(2a + 2)= 2(3a + 3)

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