Solving Linear Equations Graphically
An equation such as \(2x - 1 = 5\) can be solved with graphs: draw \(y = 2x - 1\) and \(y = 5\), and the \(x\)-coordinate where they cross is the solution. This NSW Year 9 Core maths subtopic revises and extends the Stage 4 graphing work under NESA outcome MA4‑LIN‑C‑01.
You will split an equation into two lines, one for each side, choose the right pair of lines from a list, and read the solution as the \(x\)-coordinate of the crossing point, not the \(y\)-coordinate. You will solve equations where two sloping lines cross, as in \(2x - 3 = 6 - x\), and where the crossing falls between grid lines, see why parallel lines mean an equation has no solution, compare two phone plans to find when they cost the same, and find an unknown gradient. It is part of Year 9 Core Straight Line.
Every question with a fully worked solution.
- Solving Linear Equations Graphically - Video - Intersection Point Between a Horizontal and Vertical Line Watch
Theory
To solve a linear equation graphically, graph each side as a straight line. The two lines cross at one point, and the \(x\)-coordinate of that point of intersection is the solution. This Year 9 maths page revises the earlier method, with sloping and horizontal lines, answers between grid lines, parallel lines and real cost comparisons.
An equation such as \(3x - 2 = x + 4\) asks: for which \(x\) do the two sides have the same value?
Graph the left-hand side as one line, \(y = 3x - 2\), and the right-hand side as another, \(y = x + 4\). Every point on a line satisfies its equation.
Where the lines cross, the point of intersection, both lines have the same \(y\)-value. So the \(x\)-coordinate of that point is the solution of the equation. The \(y\)-coordinate is just the value both sides equal there.
If one side is a plain number, such as \(8\), its line is horizontal: \(y = 8\).
If the two lines are parallel, they never cross, and the equation has no solution.
To solve \(\text{left side} = \text{right side}\), graph:
How to solve an equation graphically
- Split the equation: \(y = \) left side and \(y = \) right side. A plain number gives a horizontal line.
- Graph both lines on the same axes. Use a table of values or the gradient and \(y\)-intercept.
- Find the point where the lines cross.
- Read its \(x\)-coordinate. That is the solution. It may be a decimal such as \(2.5\).
- Check by substituting the solution into both sides of the original equation.
Each side of the equation becomes a line, copied exactly. The right-hand side is a number, so its line is horizontal.
| \(\text{left side}\) | \(:\) | \(y = 5x - 2\) |
| \(\text{right side}\) | \(:\) | \(y = 8\) |
The lines cross at \((2, 8)\). The solution is the \(x\)-coordinate: \(\mathbf{x = 2}\). Check: \(5 \times 2 - 2 = 8\).
The lines cross at \((3, 7)\). The solution is its \(x\)-coordinate, \(x = 3\). The \(7\) is the value both sides take there.
| \(3 \times 3 - 2\) | \(=\) | \(7\) |
| \(3 + 4\) | \(=\) | \(7 \ \checkmark\) |
The solution is \(\mathbf{x = 3}\).
Read the crossing as \(x = 3.5\), then confirm it with algebra.
| \(2x - 5\) | \(=\) | \(2\) |
| \(2x\) | \(=\) | \(7\) |
| \(x\) | \(=\) | \(\mathbf{3.5}\) |
A crossing that misses a grid corner is normal. Give it as a decimal or a fraction, \(\dfrac{7}{2}\).
Graph both lines on one set of axes and find where they cross. That solves \(4n + 15 = 2n + 35\).
| \(4n + 15\) | \(=\) | \(2n + 35\) |
| \(2n\) | \(=\) | \(20\) |
| \(n\) | \(=\) | \(\mathbf{10}\) |
After 10 visits both cost \(4 \times 10 + 15 = 55\) dollars. For fewer visits Gym A is cheaper; for more, Gym B is cheaper.
Common pitfalls
Frequently asked questions
How do you solve an equation using a graph?
Graph the left-hand side and the right-hand side as two lines on the same axes. Find where they cross. The \(x\)-coordinate of the crossing point is the solution.
Is the solution the x or the y value of the intersection?
The \(x\)-value. The \(y\)-value only tells you what both sides of the equation are equal to at the solution.
What does it mean if two lines do not intersect?
The lines are parallel, with the same gradient. The equation made from them has no solution.
What lines do you graph to solve 5x - 2 = 8?
Graph \(y = 5x - 2\) and the horizontal line \(y = 8\). They cross at \((2, 8)\), so \(x = 2\).
What if the lines cross between grid lines?
Read the \(x\)-coordinate as carefully as you can, such as \(3.5\), and confirm it by solving the equation with algebra.