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Linear Equations

Solve for x by doing the same operation to both sides until x is alone.

What Is an Equation?

An equation is like a balanced scale. Both sides must always stay equal. Whatever you do to one side, you must do exactly the same to the other side, or the balance breaks. To solve a linear equation, undo each operation surrounding x, one at a time, until x is alone on one side.

Example 1 — one step

Solve x + 4 = 9 . 1 Subtract 4 from both sides: x + 4 - 4 = 9 - 4 2 x = 5 Check: 5 + 4 = 9 ✓ Answer: x = 5

Example 2 — two steps

Solve 3 x + 2 = 14 . 1 Subtract 2 from both sides: 3 x = 12 2 Divide both sides by 3 : x = 4 Check: 3 (4) + 2 = 12 + 2 = 14 ✓ Answer: x = 4

Example 3 — x on both sides

Solve 5 x - 3 = 2 x + 9 . 1 Subtract 2 x from both sides so x only appears once: 3 x - 3 = 9 2 Add 3 to both sides: 3 x = 12 3 Divide both sides by 3 : x = 4 Check: 5 (4) - 3 = 17, and 2 (4) + 9 = 17 ✓ Answer: x = 4

Example 4 — negative coefficient

Solve 10 - 2 x = 4 . 1 Subtract 10 from both sides: - 2 x = - 6 2 Divide both sides by - 2 : x = 3 Check: 10 - 2 (3) = 10 - 6 = 4 ✓ Answer: x = 3

Practice

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Solve 4 x - 5 = 11 .

Solve 6 x + 7 = 2 x + 23 . What is x ?

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