One Step Equations (add and subtract)
This subtopic solves equations where a number has been added to or taken away from the pronumeral. In NSW Year 7 maths (NESA outcome MA4‑EQU‑C‑01) the key idea is the balance: whatever you do to one side, you do to the other.
You will use the inverse operation to get the pronumeral on its own, for example \(x + 15 = -8\) gives \(x = -23\), solve equations with the pronumeral on the right or subtracted, as in \(1 - x = 11\), and check each answer by substituting. You will also write equations for short stories, such as how many stickers Mia started with. This subtopic is part of Year 7 Algebra and Equations.
Every question with a fully worked solution.
- One Step Equations (add and subtract) - Video - How to Solve One-Step Equations (Addition and Subtraction) Watch
Theory
To solve an equation, undo what has been done to the pronumeral. Use the inverse operation and do it to both sides. This Year 7 maths page shows the balancing method and the backtracking method for equations with \(+\) and \(-\), including negative answers and word problems.
An equation is a statement with a variable, numbers and an equals sign.
The balancing method. An equation is like a balance: both sides have the same value. Do the same operation to both sides and it stays balanced.
The backtracking method. Use a flow chart to show what was done to the variable, then undo each operation with its inverse (opposite) in reverse order.
Adding and subtracting are inverse operations: \(x + 17 = 30\): subtract \(17\) from both sides. \(x - 4 = 9\): add \(4\) to both sides.
Write one step per line, then check by substituting: LHS \(=\) RHS.
Undo with the inverse:
| Equation | Do to both sides | Solution |
|---|---|---|
| \(x + 15 = 23\) | subtract \(15\) | \(x = 8\) |
| \(x - 4 = 22\) | add \(4\) | \(x = 26\) |
| \(12 = b - 5\) | add \(5\) | \(b = 17\) |
| \(8 - x = 3\) | add \(x\), then subtract \(3\) | \(x = 5\) |
How to solve a one-step equation
- Find what was done to the pronumeral (\(+\) or \(-\)).
- Do the inverse operation to both sides.
- Simplify each side, one step per line.
- Check by substituting into the original equation.
Balancing method: add \(6\) to both sides.
| \(a - 6 + 6\) | \(=\) | \(9 + 6\) |
| \(a\) | \(=\) | \(15\) |
Backtracking method: undo “\(- 6\)” by adding \(6\): \(a = 9 + 6 = 15\). Check: \(15 - 6 = 9\).
(i) \(x + 15 = 23\)
(ii) \(x - 4 = 22\)
| \(x + 15 - 15\) | \(=\) | \(23 - 15\) |
| \(x\) | \(=\) | \(8\) |
| \(x - 4 + 4\) | \(=\) | \(22 + 4\) |
| \(x\) | \(=\) | \(26\) |
(i) \(x = 8\). Check: \(8 + 15 = 23\).
(ii) \(x = 26\). Check: \(26 - 4 = 22\).
| \(x + 11 - 11\) | \(=\) | \(4 - 11\) |
| \(x\) | \(=\) | \(-7\) |
\(4 - 11\) goes \(7\) below zero. Check: \(-7 + 11 = 4\). So \(x = -7\).
| \(t - 7\) | \(=\) | \(-4\) |
| \(t - 7 + 7\) | \(=\) | \(-4 + 7\) |
| \(t\) | \(=\) | \(3\) |
It started at \(3^\circ\)C. That makes sense: it was warmer before it fell.
Add \(x\) to both sides so \(x\) is no longer subtracted: \(5 = -9 + x\). Then add \(9\) to both sides.
| \(5 + 9\) | \(=\) | \(x\) |
| \(x\) | \(=\) | \(14\) |
Check: \(5 - 14 = -9\). So \(x = 14\).
(ii) A shape has sides \(x\) cm, \(5\) cm, \(8\) cm and \(6\) cm and a perimeter of \(27\) cm. Find \(x\).
| \(m - 12\) | \(=\) | \(35\) |
| \(m\) | \(=\) | \(35 + 12 = 47\) |
| \(x + 19\) | \(=\) | \(27\) |
| \(x\) | \(=\) | \(27 - 19 = 8\) |
(i) Zoe started with \(\$47\).
(ii) \(x = 8\) cm
Common pitfalls
Frequently asked questions
What is an inverse operation?
The operation that undoes another. Adding and subtracting are inverses.
How do you solve x + 15 = 23?
Subtract \(15\) from both sides: \(x = 8\).
What is the backtracking method?
Show what was done to the variable in a flow chart, then undo each step with its inverse in reverse order.
Can the answer to an equation be negative?
Yes. \(x + 11 = 4\) gives \(x = -7\).
How do you check your answer?
Substitute it into the original equation and check that both sides are equal.