Solving Equations Graphically
Solving equations graphically means using a drawn graph as the solving tool: read the roots where a curve crosses the x-axis, meet it with a horizontal line to solve f(x) = k, and work out which straight line to draw on a curve. It is a regular Edexcel IGCSE Maths exam skill, covering linear simultaneous equations, quadratics and cubics. Every practice question below is auto-marked and comes with its own plotted graph, so scroll down and start reading solutions straight off the grid.
How to Solve Equations Graphically
A drawn graph is a solving machine. Four situations cover everything on this page; each one is a different way of reading the same graph, not a sequence of steps.
- Roots from the graph. The solutions of \(\mathrm{f}(x) = 0\) are the \(x\)-values where the graph of \(y = \mathrm{f}(x)\) crosses the \(x\)-axis.
- Solving \(\mathrm{f}(x) = k\). Draw the horizontal line \(y = k\). The solutions are the \(x\)-values where the curve meets that line.
- Solving a different equation on a drawn graph. Rearrange the target equation until the drawn function stands alone on one side. Whatever sits on the other side is the straight line to draw; the solutions are the \(x\)-values of the crossing points.
- Simultaneous equations. Draw both graphs on one grid and read the intersection points. Parallel lines never meet, so there is no solution; two equations that plot as the same line agree everywhere, so there are infinitely many.
Core ideas
Two straight lines: how many solutions?
A pair of linear simultaneous equations can behave in exactly three ways, and the graph shows which one you have.
One solution
Different gradients: the lines cross exactly once.
No solution
Same gradient, different intercepts: parallel lines never meet.
Infinitely many
Both equations plot as the same line, so every point works.
Worked examples
💡 Example 1: two lines, one solution
The graph shows \(y = 2x - 1\) and \(y = 5 - x\). Solve the simultaneous equations.
Read the crossing point from the graph: \(x = 2\) and \(y = 3\).
Check with algebra:
\[ \begin{array}{rcl} 2x - 1 &=& 5 - x \\ 3x &=& 6 \\ x &=& 2 \end{array} \]Substitute into \(y = 5 - x\):
\[ \begin{array}{rcl} y &=& 5 - 2 \\ &=& 3 \end{array} \]What's happening?
At the crossing point both equations are true at the same time, so its coordinates are the solution of the pair.
One point means one solution: the gradients are different.
💡 Example 2: roots, then f(x) = k
The graph of \(y = x^2 - 2x - 3\) is drawn. Use it to solve (a) \(x^2 - 2x - 3 = 0\) and (b) \(x^2 - 2x - 8 = 0\).
(a) The curve crosses the \(x\)-axis at \(x = -1\) and \(x = 3\).
(b) Rearrange so the drawn curve appears on the left:
\[ \begin{array}{rcl} x^2 - 2x - 8 &=& 0 \\ x^2 - 2x - 3 &=& 5 \end{array} \]Draw \(y = 5\). The curve meets it at \(x = -2\) and \(x = 4\).
What's happening?
Adding 5 to both sides rebuilds the drawn expression on the left, so the right-hand side becomes the line to draw.
The same drawn parabola solves a whole family of quadratics.
💡 Example 3: which line should be drawn?
The graph of \(y = x^2 - 3x + 1\) has been drawn. Find the line to draw to solve \(x^2 - 5x + 4 = 0\).
Add \(2x\) and subtract \(3\) on both sides to rebuild the drawn curve on the left:
\[ \begin{array}{rcl} x^2 - 5x + 4 &=& 0 \\ x^2 - 3x + 1 &=& 2x - 3 \end{array} \]So draw the line \(y = 2x - 3\) (shown dashed). The crossings give \(x = 1\) and \(x = 4\).
What's happening?
Compare the target with the drawn curve term by term: the difference is the line.
Terms swap sign as they cross the equals sign, so the drawn line takes the opposite signs of that difference.
💡 Example 4: a line meets a curve twice
The graph shows \(y = x^2 - 3\) and \(y = 2x\). Solve the simultaneous equations.
Read both crossing points: \((-1, -2)\) and \((3, 6)\).
Check with algebra:
\[ \begin{array}{rcl} x^2 - 3 &=& 2x \\ x^2 - 2x - 3 &=& 0 \\ (x-3)(x+1) &=& 0 \end{array} \]So \(x = 3\) or \(x = -1\), and the line gives \(y = 6\) or \(y = -2\).
What's happening?
A straight line can cut a parabola twice, touch it once, or miss it, so expect up to two solution pairs.
Give both crossing points: one root on its own is only half the answer.
💡 Example 5: a cubic graph
The graph of \(y = x^3\) is drawn. Find the line to draw to solve \(x^3 + x - 3 = 0\).
Rearrange so the drawn curve \(x^3\) stands alone:
\[ \begin{array}{rcl} x^3 + x - 3 &=& 0 \\ x^3 &=& 3 - x \end{array} \]So draw the line \(y = 3 - x\) (shown dashed). The single crossing gives \(x \approx 1.2\).
What's happening?
A cubic graph solves cubic equations the same way a parabola solves quadratics: rebuild the drawn curve on one side, and the other side is the line.
This cubic meets the line just once, so there is one real solution.
🔑 Key points
- Solutions of \(\mathrm{f}(x) = 0\) are the \(x\)-values where the graph crosses the \(x\)-axis.
- Solutions of \(\mathrm{f}(x) = k\) are the \(x\)-values where the graph meets the line \(y = k\).
- To solve a different equation on a drawn graph, rearrange until the drawn function is alone on one side; the other side is the line to draw.
- Simultaneous equations: the intersection points of the two graphs are the solutions.
- Parallel lines give no solution; the same line twice gives infinitely many.
- The same rearrange-and-draw method works for cubic and reciprocal graphs, not only parabolas.
⚠️ Common pitfalls
- Giving only one root when the line cuts the curve twice: read every crossing.
- Drawing \(y = \) the target equation instead of rearranging first.
- Sign slips when moving terms across: the drawn line takes the opposite sign of each moved term.
- Reading the \(x\)-value when the question asks for \(y\), or the other way round.
- Assuming the graphs always cross: a line can miss a curve, and parallel lines never meet.