An equation states that two expressions have the same value. Solving an equation means finding the value or values of the variable that make that statement true.

A linear equation in one variable can be reduced to a form such as

$$ax+b=c,$$

where the variable appears only to the first power.

Equivalent equations

An equation remains equivalent when the same quantity is added to or subtracted from both sides, or when both sides are multiplied or divided by the same nonzero quantity. These operations preserve the balance of the equation.

For example,

$$3x+5=17$$

can be transformed step by step:

$$3x=12,$$

$$x=4.$$

Substituting $x=4$ into the original equation gives $17=17$, confirming the solution.

Variables on both sides

The same idea works when the variable appears on both sides. For example,

$$5x-2=2x+7$$

becomes

$$3x=9,$$

so $x=3$.

Rearranging formulas

Equations can also express relationships between several quantities. The same operations can isolate a chosen variable. From

$$v=u+at,$$

subtracting $u$ and dividing by $a\ne0$ gives

$$t=\frac{v-u}{a}.$$

This is the same algebraic process as solving a one-variable linear equation.