Solving One-Step Equations

The one move: opposite operations

Every operation has an opposite. The opposite of addition is subtraction, and the opposite of subtraction is addition. The opposite of multiplication is division, and the opposite of division is multiplication.

Look at what's being done to the variable, then do the opposite to both sides. If the equation says $x + 8$, subtract $8$ from both sides. If it says $x - 5$, add $5$ to both sides. If it says $-4x$, divide both sides by $-4$. If it says $\dfrac{x}{3}$, multiply both sides by $3$.

After the move, one side is just the variable and the other side is the answer.

Fraction coefficients: multiply by the reciprocal

When the coefficient is a fraction, as in $\dfrac{2}{5}x = 10$, the fastest move is to multiply both sides by the reciprocal — flip the fraction. Multiplying by $\dfrac{5}{2}$ turns the left side into $1x$, because $\dfrac{5}{2} \cdot \dfrac{2}{5} = 1$. The right side becomes $10 \cdot \dfrac{5}{2} = 25$.

This is one move instead of two: you could divide by $2$ and multiply by $5$ separately, but the reciprocal does both at once.

Watch the signs

Two sign traps show up constantly. First, a negative coefficient: solving $-4x = 36$ means dividing both sides by $-4$, not $4$, so the answer is $x = -9$. Second, a small result: $x + 8 = 3$ gives $x = 3 - 8 = -5$. There's nothing wrong with a negative answer — plug it back in and check: $-5 + 8 = 3$. ✓

Worked examples

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Common questions