Adapted by Mrs. Garay

Warm Up Solve. 1. 2x + 9x – 3x + 8 = – 4 = 6x + 22 – 4x 3. + = 5 4. – = 3 x = 1 x = –13 x = x x9x 16 2x2x x = 50

Learn to solve equations with variables on both sides of the equal sign.

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.

Solve. 4x + 6 = x 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 =

Check your solution by substituting the value back into the original equation. For example, 4(  2) + 6 =  2 or  2 =  2. Helpful Hint

Solve. 9b – 6 = 5b + 18 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.

Solve. 9w + 3 = 9w + 7 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.

If the variables in an equation are eliminated and the resulting statement is false, the equation has no solution. Helpful Hint

Solve. 5x + 8 = x Your Turn! 5x + 8 = x – 5x 8 = –4x Subtract 5x from both sides. Divide both sides by –4. –2 = x 8 –4 –4x –4 =

Solve. 3b – 2 = 2b + 12 – 2b b – 2 = 12 Subtract 2b from both sides. + 2 b = 14 Add 2 to both sides. Your Turn Again!

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. One more!!!!!!!!

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.

Solve. 10z – 15 – 4z = 8 – 2z - 15 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 =

Solve. 12z – 12 – 4z = 6 – 2z + 32 Your Turn! 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 =