2 Step Equations (harder)
This subtopic stretches two-step equations with trickier signs and fractions. In NSW Year 7 maths (NESA outcome MA4‑EQU‑C‑01) it includes dividing by a negative number, a fraction line that groups a whole expression, and pronumerals on both sides.
You will decide which operation to undo first, for example multiplying by \(4\) before adding \(5\) in \(\dfrac{x - 5}{4} = -3\), and remove the smaller \(x\) term when \(x\) appears twice, as in \(7x = 3x + 10\). Answers may be zero, negative or a mixed numeral, and you check by working out each side separately. This subtopic is part of Year 7 Algebra and Equations.
Every question with a fully worked solution.
- 2 Step Equations (harder) - Video - Solving Two-Step Equations Watch
Theory
Harder two-step equations still need only two inverse operations. The skill is choosing which operation to undo first and keeping track of signs and fractions. This Year 7 maths page covers negative divisors, fraction answers, grouped tops such as \(\dfrac{x + 7}{3}\), and pronumerals on both sides.
Reverse order. In \(\dfrac{x}{4} - 6\), \(x\) was divided by \(4\), then \(6\) was taken away. Undo the \(-\,6\) first, then the \(\div 4\).
A fraction line groups. In \(\dfrac{x - 2}{6}\), the \(2\) was taken away first, then all of \(x - 2\) was divided by \(6\). So multiply by \(6\) first, then add \(2\).
A negative divisor. \(\dfrac{a}{-4}\) means \(a \div (-4)\). Undo it by multiplying both sides by \(-4\).
Pronumerals on both sides. Take the smaller \(x\) term away from both sides, then solve what is left.
Answers that are not whole numbers. Simplify the fraction, then write it as a mixed numeral or a decimal: \(\dfrac{20}{12} = \dfrac{5}{3} = 1\dfrac{2}{3}\). An answer of \(0\) is fine too: \(\dfrac{y}{2} + 3 = 3\) gives \(y = 0\).
Choose the first step carefully:
| Equation | Step 1 | Step 2 | Solution |
|---|---|---|---|
| \(\dfrac{x}{2} - 4 = 1\) | \(+\,4\) | \(\times 2\) | \(x = 10\) |
| \(\dfrac{x + 1}{3} = 5\) | \(\times 3\) | \(-\,1\) | \(x = 14\) |
| \(\dfrac{a}{-2} + 3 = 9\) | \(-\,3\) | \(\times (-2)\) | \(a = -12\) |
| \(5x = 2x + 9\) | \(-\,2x\) | \(\div 3\) | \(x = 3\) |
How to solve a harder two-step equation
- Decide which operation was done last, and undo it first.
- If \(x\) is on both sides, take away the smaller \(x\) term from both sides.
- Undo the second operation, keeping signs with their numbers.
- Simplify any fraction answer and check.
| \(\dfrac{a}{-3}\) | \(=\) | \(3 - 7 = -4\) |
| \(a\) | \(=\) | \(-4 \times (-3) = 12\) |
Check: \(\dfrac{12}{-3} + 7 = -4 + 7 = 3\). So \(a = 12\).
(i) \(3x + 8 = 2\)
(ii) \(1 - \dfrac{x}{3} = 7\)
| \(3x\) | \(=\) | \(2 - 8 = -6\) |
| \(x\) | \(=\) | \(-2\) |
| \(-\dfrac{x}{3}\) | \(=\) | \(7 - 1 = 6\) |
| \(x\) | \(=\) | \(6 \times (-3) = -18\) |
(i) \(x = -2\)
(ii) \(x = -18\). Check: \(1 - \dfrac{-18}{3} = 1 + 6 = 7\).
| \(8x\) | \(=\) | \(9 + 3 = 12\) |
| \(x\) | \(=\) | \(\dfrac{12}{8} = \dfrac{3}{2} = 1\dfrac{1}{2}\) |
\(x = 1\dfrac{1}{2}\) (or \(1.5\)). Check: \(8 \times 1.5 - 3 = 9\).
The fraction line groups \(x + 7\). Undo the \(\div 3\) first, then the \(+\,7\).
| \(x + 7\) | \(=\) | \(-2 \times 3 = -6\) |
| \(x\) | \(=\) | \(-6 - 7 = -13\) |
Check: \(\dfrac{-13 + 7}{3} = \dfrac{-6}{3} = -2\). So \(x = -13\).
Take the smaller \(x\) term, \(3x\), from both sides.
| \(14\) | \(=\) | \(4x\) |
| \(x\) | \(=\) | \(\dfrac{14}{4} = 3.5\) |
Check: LHS \(= 3 \times 3.5 + 14 = 24.5\) and RHS \(= 7 \times 3.5 = 24.5\). So \(x = 3.5\).
| \(-\dfrac{2x}{3}\) | \(=\) | \(-1 - 5 = -6\) |
| \(\dfrac{2x}{3}\) | \(=\) | \(6\) |
| \(2x\) | \(=\) | \(18\) |
| \(x\) | \(=\) | \(9\) |
Check: \(5 - \dfrac{18}{3} = 5 - 6 = -1\). So \(x = 9\).
Line 1: \(-3x = 12\)
Line 2: \(x = 4\)
Which line has the mistake?
Line 1 is right: \(21 - 9 = 12\). In line 2, divide by \(-3\), not \(3\).
| \(x\) | \(=\) | \(\dfrac{12}{-3} = -4\) |
Line 2 is wrong: \(x = -4\). Check: \(9 - 3 \times (-4) = 9 + 12 = 21\).
Common pitfalls
Frequently asked questions
How do you solve an equation with x on both sides?
Take the smaller \(x\) term away from both sides, then solve the two-step equation that is left.
How do you undo dividing by a negative number?
Multiply both sides by that negative number. \(\dfrac{a}{-3} = -4\) gives \(a = 12\).
What do you do first in (x + 7)/3 = -2?
Multiply both sides by \(3\), because all of \(x + 7\) was divided by \(3\).
Can the answer be zero?
Yes. \(\dfrac{y}{2} + 3 = 3\) gives \(y = 0\).
How do you write a fraction answer?
Simplify it, then write a mixed numeral or decimal: \(\dfrac{12}{8} = 1\dfrac{1}{2}\).