2 Step Equations
Every question with a fully worked solution.
- 2 Step Equations - Video - Solving Two-Step Equations Watch
Theory
A two-step equation needs two inverse operations to get the pronumeral on its own: undo the \(+\) or \(-\) first, then undo the \(\times\) or \(\div\). This Year 7 maths page shows the balancing and backtracking methods, negative and decimal answers, and equations from everyday stories.
Think about how the equation was built. In \(3x + 4 = 22\), \(x\) was multiplied by \(3\), then \(4\) was added.
To solve it, undo those operations in reverse order, using the inverse operation each time. Step 1. Undo adding \(4\): subtract \(4\) from both sides, giving \(3x = 18\). Step 2. Undo multiplying by \(3\): divide both sides by \(3\), giving \(x = 6\).
The balancing method does the same thing to both sides, so the equation stays balanced. The backtracking method uses a flow chart and undoes each operation in reverse order.
Check by substitution: if LHS \(=\) RHS, the answer is correct. The answer can be negative, a decimal or a fraction.
Two steps, in reverse order:
| Equation | Step 1 | Step 2 | Solution |
|---|---|---|---|
| \(5x + 1 = 21\) | \(-\,1\) | \(\div 5\) | \(x = 4\) |
| \(2x - 9 = 3\) | \(+\,9\) | \(\div 2\) | \(x = 6\) |
| \(\dfrac{x}{3} + 1 = 6\) | \(-\,1\) | \(\times 3\) | \(x = 15\) |
| \(8 - 3x = -10\) | \(-\,8\) | \(\div (-3)\) | \(x = 6\) |
How to solve a two-step equation
- Undo the \(+\) or \(-\): do the inverse to both sides.
- Undo the \(\times\) or \(\div\): do the inverse to both sides.
- Write one step per line.
- Check by substituting into the original equation.
Balancing method: subtract \(3\) from both sides, then multiply both sides by \(4\).
| \(\dfrac{x}{4} + 3 - 3\) | \(=\) | \(5 - 3\) |
| \(\dfrac{x}{4}\) | \(=\) | \(2\) |
| \(x\) | \(=\) | \(2 \times 4 = 8\) |
Backtracking method: undo “\(+ 3\)” by subtracting \(3\), then undo “\(\div 4\)” by multiplying by \(4\): \(x = (5 - 3) \times 4 = 8\). Check: \(\dfrac{8}{4} + 3 = 5\).
| \(29 - 4\) | \(=\) | \(5x\) |
| \(25\) | \(=\) | \(5x\) |
| \(x\) | \(=\) | \(\dfrac{25}{5} = 5\) |
Check: \(5 \times 5 + 4 = 29\). So \(x = 5\).
(i) \(6m - 5 = -23\)
(ii) \(\dfrac{x}{4} - 2 = 5\)
| \(6m\) | \(=\) | \(-23 + 5 = -18\) |
| \(m\) | \(=\) | \(\dfrac{-18}{6} = -3\) |
| \(\dfrac{x}{4}\) | \(=\) | \(5 + 2 = 7\) |
| \(x\) | \(=\) | \(7 \times 4 = 28\) |
(i) \(m = -3\). Check: \(6 \times (-3) - 5 = -23\).
(ii) \(x = 28\). Check: \(7 - 2 = 5\).
The \(x\) term is \(-2x\). Subtract \(7\) from both sides, then divide both sides by \(-2\).
| \(-2x\) | \(=\) | \(12 - 7 = 5\) |
| \(x\) | \(=\) | \(\dfrac{5}{-2} = -2.5\) |
Check: \(7 - 2 \times (-2.5) = 7 + 5 = 12\). So \(x = -2.5\).
| \(\dfrac{3x}{4}\) | \(=\) | \(11 - 2 = 9\) |
| \(3x\) | \(=\) | \(9 \times 4 = 36\) |
| \(x\) | \(=\) | \(\dfrac{36}{3} = 12\) |
Check: \(\dfrac{3 \times 12}{4} + 2 = 9 + 2 = 11\). So \(x = 12\).
| \(8g\) | \(=\) | \(105 - 25 = 80\) |
| \(g\) | \(=\) | \(\dfrac{80}{8} = 10\) |
Ava played \(10\) games. Check: \(8 \times 10 + 25 = 105\).
| \(6.25h + 3.5\) | \(=\) | \(28.5\) |
| \(6.25h\) | \(=\) | \(25\) |
| \(h\) | \(=\) | \(25 \div 6.25 = 4\) |
Leo hired the bike for \(4\) hours. Check: \(6.25 \times 4 + 3.50 = 28.50\).
Common pitfalls
Frequently asked questions
What is a two-step equation?
An equation that needs two inverse operations to solve, such as \(3x + 4 = 22\).
Which step do you undo first?
The \(+\) or \(-\) first, then the \(\times\) or \(\div\): the reverse of how the equation was built.
How do you solve 3x + 4 = 22?
Subtract \(4\) from both sides (\(3x = 18\)), then divide by \(3\): \(x = 6\).
Can the answer be a decimal?
Yes. \(7 - 2x = 12\) gives \(x = -2.5\).
How do you check a solution?
Substitute it into the original equation. Both sides must be equal.