Solving Equations Graphically
Solving an equation graphically means using the graph of a rule to find the \(x\)-value that makes the equation true. To solve \(2x + 1 = 7\), draw \(y = 2x + 1\) and the horizontal line \(y = 7\), then read where they meet: \(x = 3\). In NSW Year 8 maths this links equations to linear graphs (NESA outcome MA4‑LIN‑C‑01).
You will read \(y\) from a given \(x\) and \(x\) from a given \(y\), test whether a point is on the line by substituting, and choose the horizontal line that solves an equation, including answers between grid lines such as \(x = 2.5\). You will check each answer by substitution and use graphs of a growing bean plant, melting snow and the profit from a school disco. It is part of Year 8 Graphing Linear Equations.
Every question with a fully worked solution.
- Solving Equations Graphically - Video - Graphs of linear equations | Linear equations and functions Watch
Theory
Every point on a line fits its rule, so a graph can solve equations. To solve \(1 - 3x = 7\), graph \(y = 1 - 3x\), draw the horizontal line \(y = 7\), and read the \(x\)-value where they cross: \(x = -2\). This Year 8 maths page shows how.
Finding \(y\) from \(x\). Start at the \(x\)-value on the \(x\)-axis, go straight up or down to the line, then straight across to the \(y\)-axis. (Or draw the vertical line through that \(x\)-value and read where it meets the graph.)
Finding \(x\) from \(y\). Start at the \(y\)-value, go straight across to the line, then up or down to the \(x\)-axis.
Extend the line past the table points so values outside the table can still be read.
Is a point on the line? Substitute its \(x\)-value into the rule. If the rule gives its \(y\)-value, it is on the line.
Solving an equation. The left side is the rule, already drawn as a line. The right side is a number, so draw that horizontal line. Read the \(x\)-value where the two lines cross. To solve \(2x + 2 = -2\), draw \(y = -2\).
The \(x\)-axis is \(y = 0\), so where the line crosses it solves “rule \(= 0\)”. A crossing halfway between \(2\) and \(3\) gives \(x = 2.5\).
Always check by substituting the answer into the rule: a graph can only be read approximately.
Table of values for \(y = 2x + 2\):
| \(x\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|
| \(y\) | \(0\) | \(2\) | \(4\) | \(6\) |
| To solve | Draw |
|---|---|
| \(2x + 2 = -2\) | \(y = -2\) |
| \(2x + 2 = 7\) | \(y = 7\) |
| \(2x + 2 = 0\) | nothing: use the \(x\)-axis |
Solving an equation with a graph
- Graph the left side as a line \(y = \ldots\).
- Draw the horizontal line \(y = \) (right side).
- Read the \(x\)-value of the crossing point.
- Check by substituting into the rule.
Plot \((0, 1)\) and \((1, 3)\) and join them. Draw the vertical line \(x = -1\); it meets the graph at \((-1, -1)\).
| \(2 \times (-1) + 1\) | \(=\) | \(-1\) |
\(y = -1\)
Graph \(y = 1 - 3x\) through \((0, 1)\) and \((-1, 4)\). Draw \(y = 7\). The lines cross at \((-2, 7)\).
| \(1 - 3 \times (-2)\) | \(=\) | \(7\) |
\(x = -2\)
(i) Find \(y\) when \(x = 6\).
(ii) Find \(x\) when \(y = 3\).
| \(4 - 6\) | \(=\) | \(-2\) |
| \(4 - 1\) | \(=\) | \(3\) |
(i) \(y = -2\) (go down from \(6\) to the line, then across)
(ii) \(x = 1\) (go across from \(3\) to the line, then down)
(i) Which horizontal line solves \(4x + 2 = -6\)?
(ii) Solve \(4x + 2 = -6\).
The right side is \(-6\), so draw \(y = -6\). It crosses the graph at \((-2, -6)\).
| \(4 \times (-2) + 2\) | \(=\) | \(-6\) |
(i) \(y = -6\)
(ii) \(x = -2\)
(i) Use the graph to solve \(40 - 8h = 12\).
(ii) When does the lemonade run out?
| \(40 - 8 \times 3.5\) | \(=\) | \(12\) |
| \(40 - 8 \times 5\) | \(=\) | \(0\) |
(i) \(h = 3.5\) hours (halfway between \(3\) and \(4\))
(ii) After \(5\) hours, where the line meets the \(h\)-axis
(i) Solve \(2x - 6 = 3\).
(ii) Solve \(2x - 6 = 0\).
(iii) Is \((5, 5)\) on the line?
| \(2 \times 4.5 - 6\) | \(=\) | \(3\) |
| \(2 \times 3 - 6\) | \(=\) | \(0\) |
| \(2 \times 5 - 6\) | \(=\) | \(4\) |
(i) \(x = 4.5\) (\(y = 3\) meets the line halfway between \(4\) and \(5\))
(ii) \(x = 3\), where the line meets the \(x\)-axis
(iii) No: the line passes through \((5, 4)\).
Common pitfalls
Frequently asked questions
How do I solve an equation using a graph?
Graph the left side as a line, draw the horizontal line y = (right side), and read the x-value where they cross.
How do I find y for a given x on a graph?
Go from the x-value straight up or down to the line, then across to the y-axis.
What does the x-axis crossing tell me?
It solves rule = 0, because every point on the x-axis has y = 0.
How do I know if a point is on a line?
Substitute its x-value into the rule. It is on the line only if the rule gives its y-value.
Why should I check a graphical answer?
Graphs can only be read approximately, so substitute the answer into the rule.