Variables on One Side
An equation with the pronumeral on one side is solved by undoing each operation in reverse order, doing the same to both sides. This NSW Year 9 Advanced maths subtopic solves linear equations that take up to three steps, including fractions and decimals (NESA outcome MA5‑EQU‑C‑01).
You will solve equations such as \(\frac{2x - 3}{5} + 4 = 7\) and \(\frac{3 - 4x}{5} = -1.4\), and check answers by substitution. You will write and solve equations for a plumber's bill, a number puzzle, a missing test mark and a phone plan, find an unknown coefficient from a given solution, and find and correct a mistake in a student's working. It is part of Year 9 Advanced Equations.
Theory
To solve a linear equation with the pronumeral on one side, undo each operation in reverse order, doing the same to both sides. This Year 9 Advanced maths page covers inverse operations, negative and decimal coefficients, an algebraic fraction with an expression on top, word problems and finding an unknown constant, with a substitution check every time.
An equation says two expressions are equal. Solving it means finding the value of the pronumeral that makes it true. A linear equation has the pronumeral only to the power \(1\).
An equation is like a balance. Adding, subtracting, multiplying or dividing both sides by the same number (not dividing by zero) keeps it balanced.
Each operation has an inverse that undoes it: \(+\) and \(-\), \(\times\) and \(\div\). An equation is built from \(x\) by a chain of operations, so it is solved by undoing that chain in reverse order, like taking off shoes and then socks.
A term added or subtracted outside a fraction is undone before the fraction is cleared. A fraction bar acts as a bracket: in \(\dfrac{x + 2}{5}\) the whole of \(x + 2\) is divided by \(5\).
Always check by substituting the answer into the original equation.
The four balance moves, for any number \(c\):
| Equation built by | Undo with |
|---|---|
| \(+\,c\) | \(-\,c\) from both sides |
| \(-\,c\) | \(+\,c\) to both sides |
| \(\times\, c\) (including a negative \(c\)) | \(\div\, c\), sign included |
| \(\div\, c\) (a denominator) | \(\times\, c\) |
| \(c - x\) | subtract \(c\), then divide by \(-1\) |
How to solve an equation with the pronumeral on one side
- Read how the equation was built from the pronumeral: which operation came first, which came last.
- Undo the last operation first, doing the same to both sides. Usually that is a term outside a fraction, then the denominator, then the number in front of the pronumeral.
- Divide by the coefficient with its sign. Leave a fraction answer as a simplified fraction unless a decimal is asked for.
- Check by substituting into the original equation.
Word problems
- Define the pronumeral with its unit (\(k\) = kWh used, \(m\) = minutes).
- Write the equation: a fixed amount \(+\) a rate \(\times\) the pronumeral (if an allowance is included, the rate applies only to the extra: \(20 + 0.10(n - 100)\)), or the steps of a number puzzle in order ("double it, add \(5\), then divide by \(3\)" is \(\dfrac{2n + 5}{3}\)). Use the same units throughout.
- Solve, then answer the question asked, in its units (for example convert minutes to hours and minutes).
- A mean is the total divided by the number of values, so a missing value comes from \(\dfrac{\text{total} + x}{n} = \text{mean}\).
- An unknown constant: if \(x = 4\) is the solution, substitute \(x = 4\) and solve for the constant.
The \(+5\) is outside the fraction, so undo it first. Then clear the fraction.
| \(\dfrac{7 - 2x}{3}\) | \(=\) | \(2 - 5 = -3\) |
| \(7 - 2x\) | \(=\) | \(-9\) |
| \(-2x\) | \(=\) | \(-16\) |
| \(x\) | \(=\) | \(\mathbf{8}\) |
Check: \(\dfrac{7 - 16}{3} + 5 = -3 + 5 = 2\).
Move the number first, then divide by the whole coefficient, \(-0.6\), sign included.
| \(-0.6x\) | \(=\) | \(9.6 - 4.2\) |
| \(-0.6x\) | \(=\) | \(5.4\) |
| \(x\) | \(=\) | \(\dfrac{5.4}{-0.6}\) |
| \(\) | \(=\) | \(\mathbf{-9}\) |
Check: \(4.2 - 0.6(-9) = 4.2 + 5.4 = 9.6\).
Let \(k\) be the number of kWh used. Work in dollars: \(32\) cents \(= \$0.32\).
| \(90 \times 1.10 + 0.32k\) | \(=\) | \(211.80\) |
| \(99 + 0.32k\) | \(=\) | \(211.80\) |
| \(0.32k\) | \(=\) | \(112.80\) |
| \(k\) | \(=\) | \(\dfrac{112.80}{0.32}\) |
| \(\) | \(=\) | \(\mathbf{352.5}\) |
\(352.5\) kWh was used. Check: \(99 + 0.32 \times 352.5\) \(= 99 + 112.80\) \(= 211.80\).
A solution makes the equation true. Substitute \(x = 4\), then solve for \(k\).
| \(\dfrac{2(4) - k}{3}\) | \(=\) | \(5\) |
| \(8 - k\) | \(=\) | \(15\) |
| \(-k\) | \(=\) | \(7\) |
| \(k\) | \(=\) | \(\mathbf{-7}\) |
Check: with \(k = -7\) the equation is \(\dfrac{2x + 7}{3} = 5\), so \(2x + 7 = 15\) and \(x = 4\).
Common pitfalls
Frequently asked questions
How do you solve a two-step equation?
Undo the operations in reverse order. For \(3x + 4 = 19\), subtract \(4\) from both sides to get \(3x = 15\), then divide both sides by \(3\) to get \(x = 5\). Check: \(3 \times 5 + 4 = 19\).
What do you do when the x term is negative?
Divide by the negative coefficient, sign included. For \(10 - 2x = 4\), subtract \(10\) to get \(-2x = -6\), then divide by \(-2\) to get \(x = 3\).
How do you solve an equation with a fraction?
Undo anything added or subtracted outside the fraction, then multiply both sides by the denominator to clear it. Then solve the equation that is left.
How do you check the answer to an equation?
Substitute the value into the original equation and work out each side separately. If the two sides are equal, the answer is correct.
How do you turn a word problem into an equation?
Choose a letter for the unknown and write what it stands for, with units. Write each quantity in the story in terms of that letter, then use the sentence that says what they equal to write the equation.