Linear Equations

Use inverse operations to find the values that make an equation true.

The idea

An equation states that two expressions name the same number, and to solve one is to find every value of the variable that makes the statement true.

We preserve the solutions by adding or subtracting the same quantity on both sides, or by multiplying or dividing both sides by the same nonzero number. Picture a balance scale: add the same weight to both pans, or halve what sits in each pan, and it still balances. If the left pan holds $x + b$ and the right pan holds $c$, removing $b$ from both pans leaves $x$ balanced against $c - b$.

To solve, look at what has been done to the variable and undo it with the inverse operation — subtraction undoes addition, division undoes multiplication. Undo the operations in the reverse of the order they were applied, so a constant added on the outside comes off before the coefficient that multiplies the variable. When the variable stands alone on one side, the other side is the solution.

Parentheses offer a choice: multiply them out first, or divide both sides by the nonzero number in front. Both preserve the solutions, so take whichever leaves the smaller numbers.

To check a solution, substitute it into the original equation and confirm that the two sides agree.

Ways to work on it

• Walkthrough. The rules of equality on four one-variable equations.
• Practice. Solve a random ax + b = c.
• Hardest. Variable on both sides — collect like terms first.

Not sure where to start? Take the ten-question placement test.