Constructing Linear Equations
Constructing linear equations from word problems is a foundational skill assumed for Queensland Year 11 Mathematical Methods (QCAA). It turns a worded situation — a number puzzle, an age question, a perimeter or a cost — into an equation you can solve.
You will define a pronumeral, translate each phrase into algebra, form and solve the equation, then interpret the result and state its units — the backbone of every worded maths problem.
Every question with a fully worked solution.
- Constructing Linear Equations - Video - Constructing Linear Equations Watch
Theory
In Year 11 Mathematical Methods (QCAA), constructing a linear equation means turning a worded situation — a number puzzle, an ages problem, a perimeter, or a cost — into an equation you can solve. This page shows how to define a pronumeral, form the equation from the words, solve it one step per line, and interpret the answer in context.
To construct a linear equation, first define a pronumeral for the unknown quantity (for example, let \(x\) be the number, or \(w\) the width in metres). Then translate each phrase in the problem into algebra: ‘trebled’ means \(\times 3\), ‘decreased by \(5\)’ means \(-5\), ‘the sum is’ means \(=\).
Writing an expression for each described quantity lets you form an equation by setting two expressions equal. You then solve it as an ordinary linear equation and interpret the result — a length, an age, or a cost — back in the words of the question.
Common phrases translate straight into algebra. If \(x\) is the number:
A rectangle of length \(l\) and width \(w\) has perimeter
A price of \(x\) with \(10\%\) GST added costs
How to construct and solve a linear equation
- Define a pronumeral for the unknown, stating its meaning and unit.
- Translate each sentence into an expression, and set two expressions equal to form the equation.
- Solve the linear equation one operation per line.
- Interpret the solution back in context, and check it fits the words.
Define — let \(n\) be the number. Translate the words:
| \(3n-5\) | \(=\) | \(34\) |
Solve — add \(5\), then divide by \(3\):
| \(3n\) | \(=\) | \(34+5\) |
| \(3n\) | \(=\) | \(39\) |
| \(n\) | \(=\) | \(13\) |
Check — treble and decrease by \(5\):
| \(3(13)-5\) | \(=\) | \(39-5\) |
| \(=\) | \(34\;\checkmark\) |
The number is \(13\).
Define — let \(x\) be the brother’s age; Mia is \(x+4\). Sum is \(28\):
| \(x+(x+4)\) | \(=\) | \(28\) |
Solve — collect like terms, then solve:
| \(2x+4\) | \(=\) | \(28\) |
| \(2x\) | \(=\) | \(24\) |
| \(x\) | \(=\) | \(12\) |
Brother \(=12\), so Mia \(=12+4=16\).
Check — the ages sum to:
| \(12+16\) | \(=\) | \(28\;\checkmark\) |
The brother is \(12\) and Mia is \(16\).
Define — let \(w\) be the width (m); length \(=w+3\). Use \(P=2(l+w)\):
| \(2\big((w+3)+w\big)\) | \(=\) | \(26\) |
Solve — simplify inside, expand, then solve:
| \(2(2w+3)\) | \(=\) | \(26\) |
| \(4w+6\) | \(=\) | \(26\) |
| \(4w\) | \(=\) | \(20\) |
| \(w\) | \(=\) | \(5\) |
Width \(=5\) m, so length \(=5+3=8\) m.
Check — the perimeter is:
| \(2(8+5)\) | \(=\) | \(2(13)=26\;\checkmark\) |
Width \(5\) m and length \(8\) m.
Define — let \(x\) be the amount before GST (dollars). Adding \(10\%\) gives:
| \(x+0.1x\) | \(=\) | \(319\) |
| \(1.1x\) | \(=\) | \(319\) |
Solve — divide both sides by \(1.1\):
| \(x\) | \(=\) | \(\dfrac{319}{1.1}\) |
| \(x\) | \(=\) | \(290\) |
Check — add \(10\%\) back:
| \(1.1\times 290\) | \(=\) | \(319\;\checkmark\) |
The amount before GST is \(\$290\).
Common pitfalls
Frequently asked questions
How do you turn a word problem into a linear equation?
Define a pronumeral for the unknown, write each described quantity as an expression in that pronumeral, then set two expressions equal to form the equation.
What does 'trebled and decreased by 5' become in algebra?
If the number is \(n\), it becomes \(3n-5\): treble means \(\times 3\), and decreased by \(5\) means \(-5\).
How do you set up an ages problem?
Let a pronumeral be one person's age and write the other ages in terms of it, then use the given relationship. If a brother is \(x\) and Mia is \(4\) older, Mia is \(x+4\).
How do you write a perimeter equation for a rectangle?
Use \(P=2(l+w)\). If the length is \(3\) more than the width \(w\), the perimeter is \(2\big((w+3)+w\big)\).
How do you find a price before 10% GST?
The total including GST is \(1.1x\), so set \(1.1x\) equal to the total and divide by \(1.1\). For \(\$319\) the pre-GST amount is \(\$290\).
Should I always state units in the answer?
Yes. Give the final answer as a sentence with units, such as ‘the width is \(5\) m’, so it answers the question that was asked.