In Grade 6 you solved one-step equations like $4x = 20$. The idea is the same now, but the numbers can be negative, fractions or decimals, so you need the signed-number rules too.
Key idea
Undo what was done to $x$ by doing the inverse operation to both sides. Divide by the coefficient itself, sign and all. For a fraction coefficient, multiply both sides by its reciprocal.
Why it works
An equation is a balance: both sides have the same value. Doing the same thing to both sides keeps them equal. To get $x$ alone after it was multiplied by $-6$, divide by $-6$, because $\frac{-6x}{-6} = x$. For $-\frac{2}{3}x$, multiply by $-\frac{3}{2}$, because $-\frac{3}{2} \times \left(-\frac{2}{3}\right) = 1$.
Worked examples
Worked example: Adding and subtracting decimals
Solve $x - 4.5 = -2$.
- Undo subtracting 4.5: add 4.5 to both sides.
- $x = -2 + 4.5 = 2.5$.
- Check: $2.5 - 4.5 = -2$ ✓.
Answer: $x = 2.5$
Worked example: A negative coefficient
Solve $-6x = 42$.
- $x$ is multiplied by $-6$. Divide both sides by $-6$.
- $x = 42 \div (-6) = -7$.
- Check: $-6 \times (-7) = 42$ ✓.
Answer: $x = -7$
Worked example: A fraction coefficient
Solve $-\frac{2}{3}x = 8$.
- Multiply both sides by the reciprocal, $-\frac{3}{2}$.
- $x = 8 \times \left(-\frac{3}{2}\right) = -12$.
- Check: $-\frac{2}{3} \times (-12) = 8$ ✓.
Answer: $x = -12$
Common mistake
Dropping the sign of the coefficient (dividing $-6x = 42$ by $6$ gives $x = 7$) or dividing by the opposite number. Always check by putting your answer back into the original equation: $-6 \times 7 = -42$, not $42$.