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Solving Equations with a Variable on Both Sides

Sometimes, the unknown quantity will appear on both sides of an equation. This is where the properties learned in 5.1 and 5.2 come in handy. A quantity with a variable can be treated just like a quantity without variables -- a quantity with a variable follows all the rules learned in the last two sections. For example, we can add a quantity with a variable to both sides without changing the equation or the values that make it true:

15 - x = 4x
15 - x + x = 4x + x
15 + 0x = 5x
15 = 5x
3 = x
x = 3

Similarly, we can subtract a term with a variable from both sides of the equation:
5x = 6 + 2x
5x - 2x = 6 + 2x - 2x
3x = 6 + 0x
3x = 6
x = 2

After simplifying, the first step in solving an equation with a variable on both sides is to get the variable on one side. This is done by reversing the addition or subtraction of one of the terms with the variable. In other words, we must add to both sides or subtract from both sides one of the quantities that contains the variable. It is generally easier to add or to subtract the smaller quantity from the larger quantity, so we are working with positive coefficients, but either way works. Once the variable is on one side only, we can proceed using inverse operations, as in 4.1 and 4.2.

Example 1. Solve for x: 3x + 2x = 12 - x

  • Simplify: 5x = 12 - x
  • Get the variable on one side:
    • 5x + x = 12 - x + x
    • 6x = 12
  • Solve using inverse operations:
    • =
    • x = 2
  • Check: 3(2) + 2(2) = 12 - 2 ? Yes!

Example 2. Solve for y: 5y - 3 = 3y + 5
  • The equation is already simplified.
  • Get the variable on one side:
    • 5y - 3 - 3y = 3y + 5 - 3y
    • 5y - 3y - 3 = 3y - 3y + 5
    • 2y - 3 = 5
  • Solve using inverse operations:
    • 2y - 3 + 3 = 5 + 3
    • 2y = 8
    • =
    • y = 4
  • Check: 5(4) - 3 = 3(4) + 5 ? Yes!

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