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# Solving Equations with Variables on Both Sides

Solving Equations with Variables on Both Sides. Lesson 2-4. Many equations contain variables on each side. To solve these equations, FIRST use addition and subtraction to write an equivalent equation with all of the variables on one side.

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## Solving Equations with Variables on Both Sides

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1. Many equations contain variables on each side. To solve these equations, FIRST use addition and subtraction to write an equivalent equation with all of the variables on one side. Traditionally, the variables (letter) are “moved to the ___________________ and the constants (numbers) are “moved” to the ________________ side of the equal sign. left right

2. 8 + 5s = 7s – 2 _________________________________ subtract “7s” so variables are on the left. _________________________________ Simplify ___________________________________ Subtract “8” so constants are on the right. ___________________________________ Simplify ___________________________________ Multiply/Divide ___________________________________ Simplify CHECK: __________________________________ ___________________________________ ____________________________________ 8 + 5s – 7s = 7s – 7s – 2 8 – 2s = – 2 8 – 8 – 2s = – 2 – 8 – 2s = – 10 s = 5 Substitute the value of the variable for the variable 8 + 5(5) = 7(5) - 2 8 + 25 35 - 2 33 33

3. Multiply both sides by the LCD 12 – r = 2r + 3 Simplify using the distributive property. 12 – r – 2r = 2r – 2r+ 3Subtract 2r from each side, this “moves” the variable to the left side of the equation. 12 – 3r = 3 -r -2r = -3r 12 – 12 – 3r = 3 – 12Subtract 12 from each side, this “moves” the constant to the right side of the equation. – 3r = – 9 Simplify “3 – 12” Divide each side by -3 (coefficient of r) r = 3 Simplify

4. CHECK: 3 – 0.75 1.5 + 0.75 2.25 2.25 Since the two sides are the same, the equation checks.

5. Practice solving the problems below. Answers are on the last slide of the part 2 notes.

6. Solving Equations with Grouping Symbols 6 + 4q = 12q - 42 6 + 4q – 12q = 12q – 12q - 42 6 – 8q = - 42 6 – 6 – 8q = – 42 – 6 – 8q = – 48 q = 6 CHECK: 30 = 30

7. Yes, there is a “typo” in your notes. 18r – 24 = 21r + 3 Use the distributive property. 18r – 24 – 21r = 21r – 21r + 3 Subtract 21r from each side to eliminate the variables on the right side.. Simplify by combining Like Terms (18r and -21r). – 3r – 24= 3 – 3r – 24 + 24 = 3 + 24 Add 24 to each side to eliminate the constants on the left side. – 3r = 27 Simplify (add 3 + 24) Divide each side by -3 (the coefficient of r.) r = – 9 Simplify

8. Check:

9. Practice solving the problems below. Answers are on the last slide of the part 2 notes.

10. Please check back for part 2 of these notes where we investigate equations that have no solution or infinitely many solutions.

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