One Step Equations
A one-step equation needs a single inverse operation to find the unknown: undo adding by subtracting, undo multiplying by dividing, and always do the same to both sides. In NSW Year 8 maths (NESA outcome MA4‑EQU‑C‑01) the numbers get harder, with negatives, decimals and fractions, as in \(-3x = -24\) or \(0.4x = 6\).
You will solve equations such as \(x - 11 = 20\), \(\frac{c}{3} = -15\) and \(\frac{t}{2} = \frac{3}{4}\), deal with a subtracted pronumeral as in \(3 - x = 5\), and check each answer by substituting it back in. You will also write an equation from a short story, such as finding the cost of one calculator when 12 of them cost $174. It is part of Year 8 Equations.
Theory
To solve a one-step equation, apply the inverse operation to both sides. The inverse of adding is subtracting, and the inverse of multiplying is dividing: \(x + 5 = 9\) gives \(x = 9 - 5 = 4\), and \(\dfrac{x}{2} = 8\) gives \(x = 2 \times 8 = 16\). This Year 8 maths page shows how.
An expression such as \(4n\) has no equals sign. An equation such as \(4n = 20\) does. Solving it means finding the value of the pronumeral that makes both sides equal.
Inverse operations undo each other: \(+\) and \(-\); \(\times\) and \(\div\). Whatever you do to one side, do to the other, so the equation stays balanced. Write one step per line.
Keep the sign with the number. In \(-8x = 24\), divide both sides by \(-8\): \(x = -3\). In \(\dfrac{-x}{5} = 2\), multiply both sides by \(-5\): \(x = -10\).
The number can come first. \(6 + x\) still means \(x\) with \(6\) added, so subtract \(6\). In \(6 - x\) the pronumeral is subtracted: add \(x\) to both sides first, then undo the number, and write the answer with the pronumeral on the left.
Decimals and fractions. A decimal in front is still a multiplier: \(0.5x = 3\) gives \(x = 6\). Answers need not be whole: \(\dfrac{n}{5} = \dfrac{2}{3}\) gives \(n = \dfrac{10}{3} = 3\dfrac{1}{3}\). When two things are done to the pronumeral, undo them in reverse order: \(\dfrac{5m}{2} = 20\) gives \(5m = 40\), so \(m = 8\).
Check by substituting: the left-hand side (LHS) must equal the right-hand side (RHS). Word problems: name the pronumeral and its unit, write the equation, solve it and answer in words.
Inverse operations:
| Operation on the pronumeral | Inverse (do to both sides) | Example |
|---|---|---|
| add | subtract | \(x + 5 = 9\), so \(x = 4\) |
| subtract | add | \(x - 5 = 9\), so \(x = 14\) |
| multiply | divide | \(2x = 8\), so \(x = 4\) |
| divide | multiply | \(\dfrac{x}{2} = 8\), so \(x = 16\) |
How to solve a one-step equation
- See what has been done to the pronumeral.
- Do the inverse operation to both sides.
- Check: substitute and show LHS = RHS.
(i) \(x + 5 = 9\)
(ii) \(x - 5 = 9\)
| \(x + 5 - 5\) | \(=\) | \(9 - 5\) |
| \(x\) | \(=\) | \(4\) |
| \(x - 5 + 5\) | \(=\) | \(9 + 5\) |
| \(x\) | \(=\) | \(14\) |
(i) \(x = 4\)
(ii) \(x = 14\)
(i) \(2x = 8\)
(ii) \(\dfrac{x}{2} = 8\)
| \(\dfrac{2x}{2}\) | \(=\) | \(\dfrac{8}{2}\) |
| \(x\) | \(=\) | \(4\) |
| \(2 \times \dfrac{x}{2}\) | \(=\) | \(2 \times 8\) |
| \(x\) | \(=\) | \(16\) |
(i) \(x = 4\)
(ii) \(x = 16\)
(i) \(p - 13 = -5\)
(ii) \(9k = -108\)
| \(p\) | \(=\) | \(-5 + 13 = 8\) |
| \(k\) | \(=\) | \(\dfrac{-108}{9} = -12\) |
(i) \(p = 8\). Check: \(8 - 13 = -5\).
(ii) \(k = -12\). Check: \(9 \times (-12) = -108\).
(i) \(m + 6.4 = 2.9\)
(ii) \(\dfrac{q}{4} = -2.5\)
| \(m\) | \(=\) | \(2.9 - 6.4 = -3.5\) |
| \(q\) | \(=\) | \(-2.5 \times 4 = -10\) |
(i) \(m = -3.5\)
(ii) \(q = -10\)
(i) \(12 - x = 19\)
(ii) \(\dfrac{-y}{4} = 3\)
(i) Add \(x\) to both sides: \(12 = 19 + x\). Subtract \(19\): \(-7 = x\).
(ii) Multiply both sides by \(-4\).
| \(x\) | \(=\) | \(-7\) |
| \(y\) | \(=\) | \(3 \times (-4) = -12\) |
(i) \(x = -7\). Check: \(12 - (-7) = 19\).
(ii) \(y = -12\). Check: \(\dfrac{-(-12)}{4} = 3\).
Let \(w\) be Jack’s weekly pay in dollars.
| \(\dfrac{w}{4}\) | \(=\) | \(63.75\) |
| \(w\) | \(=\) | \(63.75 \times 4 = 255\) |
Jack’s weekly pay is \(\$255.00\). Check: \(255 \div 4 = 63.75\).
Common pitfalls
Frequently asked questions
What is an inverse operation?
An operation that undoes another: subtracting undoes adding, dividing undoes multiplying.
How do you solve x + 7 = 12?
Subtract \(7\) from both sides: \(x = 5\).
How do you solve x divided by 3 equals 5?
Multiply both sides by \(3\): \(x = 15\).
How do you check a solution?
Substitute it into the equation; the left-hand side must equal the right-hand side.
What is the difference between an expression and an equation?
An equation has an equals sign, such as \(2x = 10\); an expression, such as \(2x\), does not.