One-Step Equations

An equation is a balance. Whatever you do to one side, you must do to the other, or it stops being balanced.

To get x alone, undo whatever is being done to it using the opposite operation. Addition undoes subtraction. Multiplication undoes division.

x + 7 = 12 → subtract 7 from both sides → x = 5.
4x = 20 → divide both sides by 4 → x = 5.
x/3 = 6 → multiply both sides by 3 → x = 18.

Always check by substituting your answer back in. It takes five seconds and catches nearly everything.

Solving relies on the properties of equality: for any c, if a = b then a + c = b + c and ac = bc (with c ≠ 0 for division). These guarantee the solution set is unchanged, which is what "equivalent equations" means.

The operations used to isolate a variable are inverses in the group-theoretic sense: adding c is undone by adding −c, multiplying by c is undone by multiplying by c⁻¹. That is why "do the opposite" is a reliable instruction rather than a heuristic.

The c ≠ 0 restriction on multiplication is not pedantry. Multiplying both sides by zero turns any equation into 0 = 0, destroying information, and dividing both sides by an expression that might be zero can delete valid solutions — an error that appears in earnest when solving rational and quadratic equations.