Equations with the Unknown on Both Sides

A letter on both sides

In 5x + 3 = 2x + 12 there is an x term on each side of the equation. Subtracting 3 or dividing by 5 does not help: x would still be on both sides. The first job is to get the x terms onto one side only.

Take 2x off both pans. The scales stay level, the left keeps 3x + 3, and the right keeps 12.

Take the smaller x term off both sides

An equation is a level balance, so the same amount can come off both pans. Subtract 2x from both sides: 5x − 2x = 3x on the left, and 2x − 2x = 0 on the right. The equation becomes 3x + 3 = 12, with x on the left only.

Take off the smaller x term, 2x, not the larger one. Subtracting 5x from both sides would also work, but it leaves 3 = −3x + 12, with a negative x term to deal with. Removing the smaller term keeps the x term positive.

Then finish as usual

3x + 3 = 12 is the kind of equation you already solve. Subtract 3 from both sides: 3x = 9. Divide both sides by 3: x = 3.

Check the answer in the equation you started with. The left side is 5 × 3 + 3 = 18, and the right side is 2 × 3 + 12 = 18. Both sides are 18, so x = 3 is right.

What the answer means

Each side of an equation is an expression whose value changes as x changes. The solution is the value of x that makes the two sides the same number. For 5x + 4 = 2x + 19, subtract 2x from both sides: 3x + 4 = 19. Subtract 4: 3x = 15. Divide by 3: x = 5.

At x = 5 the left side is 5 × 5 + 4 = 29 and the right side is 2 × 5 + 19 = 29. At any other value of x the two sides are different numbers.

Slide x along the axis and read both sides of 5x + 4 = 2x + 19. Find the x where they agree.

When an x term is subtracted

In 2x + 1 = 7 − x, the right side has −x, one x taken away. Add x to both sides to clear it: 2x + x + 1 = 7, so 3x + 1 = 7. Subtract 1 from both sides: 3x = 6. Divide by 3: x = 2. Check: 2 × 2 + 1 = 5 and 7 − 2 = 5.

The usual mistakes

Moving a term without changing its sign. In 5x + 3 = 2x + 12, the 2x is subtracted from both sides, so the left becomes 5x − 2x = 3x. Writing 5x + 2x = 7x on the left adds it instead, and the answer comes out wrong.

Taking the x term off one side only. Subtracting 2x from the right and not from the left tips the balance, and the new equation is no longer true.