What you'll learn
Solving equations graphically means finding solutions by reading where graphs cross. In this guide you will learn how to solve linear and quadratic equations using their graphs, how to find the intersection of two graphs, how to use a graph to solve a related equation, and how to interpret these solutions. This connects algebra and graphs at GCSE.
Key terms and definitions
Intersection — the point(s) where two graphs cross.
Root / solution — an x-value that satisfies an equation; graphically, where a curve meets the x-axis or another graph.
Simultaneous solution — the (x, y) values satisfying two equations at once.
Line of the equation — the graph you draw to solve an equation.
Reading off — finding values from a graph using the scales.
Core concepts
Solving an equation from a single graph
The solutions of f(x) = 0 are where the graph of y = f(x) crosses the x-axis (the roots). For example, the solutions of x² − x − 6 = 0 are the x-values where y = x² − x − 6 cuts the x-axis (x = −2 and x = 3). Read these from the graph.
Intersection of two graphs
The point(s) of intersection of two graphs give the simultaneous solution of their equations — the x and y values that satisfy both. For example, where y = x + 1 meets y = x² − 1, the x-coordinates are the solutions of x + 1 = x² − 1.
Solving a related equation
You can use a graph you already have to solve a different but related equation. To solve, say, x² − x − 6 = 2 using the graph of y = x² − x − 6, draw the line y = 2 and read off the x-values where it crosses the curve. More generally, rearrange the target equation so one side matches the drawn curve, then draw the line for the other side.
Reading solutions accurately
Solutions read from a graph are usually approximate, limited by the scale. Use the scales carefully, read to a sensible accuracy (e.g. one decimal place), and give all solutions. A quadratic graph often gives two intersection points with a line.
Interpreting in context
In real-life graphs, an intersection might represent when two quantities are equal — for example, when two phone tariffs cost the same, or when two vehicles meet. The coordinates then have a practical meaning.
Worked examples
Example 1: Roots from a graph
The graph of y = x² − 4 crosses the x-axis at x = −2 and x = 2. What equation does this solve?
These are the solutions of x² − 4 = 0, namely x = −2 and x = 2.
Example 2: Intersection
Two graphs y = x² and y = x + 2 intersect. What equation do the x-coordinates solve?
Setting them equal: x² = x + 2, i.e. x² − x − 2 = 0, so x = −1 and x = 2.
Example 3: Related equation
Using the graph of y = x² − 3, how would you solve x² − 3 = 1?
Draw the horizontal line y = 1 and read off the x-values where it crosses the curve.
Common mistakes and how to avoid them
Reading the wrong axis. Solutions of f(x) = 0 are the x-values where the curve meets the x-axis.
Missing a solution. A curve and line often cross twice — give both.
Not drawing the right line. To solve a related equation, draw the line matching the other side.
Over-precise answers. Graphical solutions are approximate; read to a sensible accuracy.
Misreading scales. Check what each square represents.
Exam technique for Solving Equations Graphically
Identify the x-axis crossings for f(x) = 0.
Find intersections of two graphs for simultaneous solutions.
Draw the correct extra line to solve a related equation.
Read all solutions, usually two for a quadratic.
Use the scales and give answers to a sensible accuracy.
Quick revision summary
To solve equations graphically, use where graphs cross. The solutions of f(x) = 0 are where y = f(x) crosses the x-axis (its roots). The intersection of two graphs gives the simultaneous solution of their equations — set the two expressions equal to see which equation the x-coordinates solve (y = x² and y = x + 2 meet where x² − x − 2 = 0). To solve a related equation using an existing curve, draw the line matching the other side (to solve x² − 3 = 1 with y = x² − 3, draw y = 1) and read off the x-values where they cross. Graphical solutions are approximate, so use the scales carefully and read to a sensible accuracy, giving all solutions (often two for a quadratic). The common errors are reading the wrong axis, missing a second solution, drawing the wrong line, and over-precision. Find x-axis crossings for f(x) = 0, use intersections for simultaneous equations, draw the correct extra line for related equations, and read all solutions from the scales.