One Step Equations (multiply and divide)
Every question with a fully worked solution.
- One Step Equations (multiply and divide) - Video - How to Solve One-Step Equations (Multiplication and Division) Watch
Theory
Multiplying and dividing undo each other. Divide both sides to undo a multiplication, and multiply both sides to undo a division. This Year 7 maths page shows how to solve equations such as \(2x = 32\) and \(\dfrac{x}{7} = 6\), with negative numbers, decimal answers and two steps.
\(3x\) means \(3 \times x\), and \(\dfrac{m}{4}\) means \(m \div 4\). To get the pronumeral on its own, do the inverse operation to both sides.
\(5x = 45\): undo multiplying by \(5\), so divide both sides by \(5\): \(\dfrac{5x}{5} = \dfrac{45}{5}\), so \(x = 9\).
\(\dfrac{n}{6} = 4\): undo dividing by \(6\), so multiply both sides by \(6\): \(\dfrac{n}{6} \times 6 = 4 \times 6\), so \(n = 24\).
Keep the sign with the number. For \(-4x = 20\), divide both sides by \(-4\). For \(\dfrac{a}{-2} = 7\), multiply both sides by \(-2\), so \(a = -14\). For \(\dfrac{-x}{3} = 4\), multiply both sides by \(-3\), so \(x = -12\).
The answer does not have to be a whole number: \(4x = 14\) gives \(x = 14 \div 4 = 3.5\).
Undo with the inverse:
| Equation | Do to both sides | Solution |
|---|---|---|
| \(2x = 32\) | divide by \(2\) | \(x = 16\) |
| \(\dfrac{x}{7} = 6\) | multiply by \(7\) | \(x = 42\) |
| \(-5x = 35\) | divide by \(-5\) | \(x = -7\) |
| \(\dfrac{a}{-2} = 7\) | multiply by \(-2\) | \(a = -14\) |
How to solve a multiply or divide equation
- Find what was done to the pronumeral (\(\times\) or \(\div\)), with its sign.
- Do the inverse to both sides (\(\div\) undoes \(\times\); \(\times\) undoes \(\div\)).
- Simplify, one step per line.
- Check by substituting.
Balancing method: multiply both sides by \(7\).
| \(\dfrac{x}{7} \times 7\) | \(=\) | \(6 \times 7\) |
| \(x\) | \(=\) | \(42\) |
Backtracking method: undo “\(\div 7\)” by multiplying by \(7\): \(x = 6 \times 7 = 42\). Check: \(\dfrac{42}{7} = 6\).
(i) \(2x = 32\)
(ii) \(\dfrac{y}{5} = 9\)
| \(\dfrac{2x}{2}\) | \(=\) | \(\dfrac{32}{2}\) |
| \(x\) | \(=\) | \(16\) |
| \(\dfrac{y}{5} \times 5\) | \(=\) | \(9 \times 5\) |
| \(y\) | \(=\) | \(45\) |
(i) \(x = 16\)
(ii) \(y = 45\)
(i) \(\dfrac{m}{7} = -4\)
(ii) \(-5x = 35\)
| \(m\) | \(=\) | \(-4 \times 7 = -28\) |
| \(x\) | \(=\) | \(\dfrac{35}{-5} = -7\) |
(i) \(m = -28\). Check: \(\dfrac{-28}{7} = -4\).
(ii) \(x = -7\). Check: \(-5 \times (-7) = 35\).
(i) \(\dfrac{a}{-6} = 5\)
(ii) \(\dfrac{-x}{4} = -3\)
| \(a\) | \(=\) | \(5 \times (-6) = -30\) |
| \(-x\) | \(=\) | \(-3 \times 4 = -12\) |
| \(x\) | \(=\) | \(12\) |
(i) \(a = -30\)
(ii) \(x = 12\). Check: \(\dfrac{-12}{4} = -3\).
| \(8c\) | \(=\) | \(124\) |
| \(c\) | \(=\) | \(\dfrac{124}{8} = 15.5\) |
One ball costs \(\$15.50\). The answer does not have to be a whole number. Check: \(8 \times 15.5 = 124\).
| \(\dfrac{3x}{4} \times 4\) | \(=\) | \(-9 \times 4\) |
| \(3x\) | \(=\) | \(-36\) |
| \(x\) | \(=\) | \(\dfrac{-36}{3} = -12\) |
Check: \(\dfrac{3 \times (-12)}{4} = \dfrac{-36}{4} = -9\). So \(x = -12\).
| \(\dfrac{b}{3}\) | \(=\) | \(27\) |
| \(b\) | \(=\) | \(27 \times 3 = 81\) |
The bill was \(\$81\).
Common pitfalls
Frequently asked questions
How do you solve 2x = 32?
Divide both sides by \(2\): \(x = 16\).
How do you solve x/7 = 6?
Multiply both sides by \(7\): \(x = 42\).
How do you solve -5x = 35?
Divide both sides by \(-5\): \(x = -7\).
Does the answer have to be a whole number?
No. \(4x = 14\) gives \(x = 3.5\).
Which operation do you undo first?
The one that was done last. Backtracking undoes the steps in reverse order.