Solving an equation graphically means drawing the graphs of two functions and reading off the x-value where they intersect. At KS3 this is most often used to solve equations like x² = 5 or x³ − 2x = 1 that are difficult to solve by algebraic rearrangement alone.
Why solve equations with graphs?
Some equations have solutions that cannot be expressed exactly as fractions or surds. Others have multiple solutions that are hard to find algebraically. A graphical approach finds approximate solutions quickly and visually — you can see how many solutions exist and roughly where they are.
The trade-off is precision: reading from a grid gives solutions to 1 or 2 decimal places at best, whereas algebra can give exact values. Use graphs when the question asks for approximate or graphical solutions, or when the algebra is too complex for the level.
The core method: two graphs intersecting
Every graphical equation-solving problem follows the same structure:
- Rearrange the equation into the form f(x) = g(x)
- Draw the graph of y = f(x) and the graph of y = g(x) on the same axes
- Find the x-coordinates where the graphs intersect — these are the solutions
Alternatively, rearrange to f(x) = 0 and draw y = f(x), then read off where it crosses the x-axis.
Method 1: Using y = 0 (finding x-intercepts)
Example: Solve x² − x − 2 = 0 graphically.
Draw y = x² − x − 2.
Build a table of values:
| x | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| x² | 4 | 1 | 0 | 1 | 4 | 9 |
| −x | 2 | 1 | 0 | −1 | −2 | −3 |
| −2 | −2 | −2 | −2 | −2 | −2 | −2 |
| y | 4 | 0 | −2 | −2 | 0 | 4 |
Plot the points and draw a smooth curve. The curve crosses y = 0 (the x-axis) at x = −1 and x = 2.
Solutions: x = −1 and x = 2. (These can be verified algebraically: x² − x − 2 = (x + 1)(x − 2) = 0.)
Method 2: Using two graphs to intersect
Example: Solve x² = 3x − 1 graphically, giving solutions to 1 decimal place.
Rather than rearranging to y = 0, draw y = x² and y = 3x − 1 separately. Where they cross, x² = 3x − 1.
Table of values for both:
| x | 0 | 1 | 2 | 3 | 0.5 | 2.5 |
|---|---|---|---|---|---|---|
| y = x² | 0 | 1 | 4 | 9 | 0.25 | 6.25 |
| y = 3x − 1 | −1 | 2 | 5 | 8 | 0.5 | 6.5 |
Plot both curves on the same grid. They intersect near x ≈ 0.4 and x ≈ 2.6.
Approximate solutions: x ≈ 0.4 and x ≈ 2.6.
How do you use an existing graph to solve a new equation?
Exam questions often draw a graph for you (e.g. y = x² + x) and then ask you to solve a new equation using it. The trick is to rearrange the new equation until one side matches the given graph's function.
Example: The graph of y = x² + x is drawn. Use it to solve x² + x − 3 = 0.
Rearrange: x² + x = 3.
So you need the x-values where y = x² + x meets y = 3. Draw the horizontal line y = 3 on the existing graph and read off the x-coordinates of the intersection points.
This technique saves drawing a new curve — you just add a straight line to the existing diagram.
How many solutions can a graphical equation have?
| Equation type | Typical number of solutions |
|---|---|
| Linear: ax + b = 0 | Exactly 1 (unless a = 0) |
| Quadratic: ax² + bx + c = 0 | 0, 1, or 2 |
| Cubic: ax³ + ... = 0 | 1, 2, or 3 |
A quadratic graph is a parabola. It meets y = 0 in two places (two solutions), one place (one repeated solution, touching the axis), or not at all (no real solutions). Looking at the graph immediately tells you how many solutions exist.
Accuracy and presentation tips
- Use a sharpened pencil and a ruler for straight lines; draw curves freehand in a smooth, continuous motion.
- Label both axes and mark the scale clearly.
- Plot at least six to eight points for a quadratic; more for a cubic.
- Draw a small circle at each intersection point and mark its x-coordinate clearly.
- Graphical solutions are approximate: state "x ≈ ..." rather than "x =..." unless the intersection falls exactly on a grid line.
Frequently asked questions
How many decimal places should my graphical answer be given to?
Usually to 1 decimal place, or to the accuracy the grid allows. The question often specifies: "Give your answers to 1 d.p." If reading between grid lines, estimate carefully. If the intersection falls exactly on a grid point, state the exact value.
What if the graphs don't intersect?
If the graphs do not cross, the equation has no real solution. For example, if you draw y = x² and y = −1 (a horizontal line below the x-axis), they never meet — meaning x² = −1 has no real solutions. State "no real solution" or "the graphs do not intersect."
Can I use trial and improvement instead of a graph?
Trial and improvement is a numerical method that converges on a solution without drawing a graph. Both methods give approximate answers. Graphical methods show all solutions simultaneously and their approximate locations; trial and improvement finds one solution at a time to any desired accuracy. Questions will specify which method to use.
Do I need to draw the graph for every question, or can I read from an existing one?
Often the question provides a pre-drawn graph and asks you to use it. In that case, draw the additional line(s) needed and read off the intersections — you do not redraw the given curve. Only draw from scratch when the question says "by drawing a graph" or "draw the graph of ... and hence solve ..."
Practise graphical equation-solving step by step with Professor Pi at aitutors.me.