Solving Equations Graphically — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers using a drawn graph to solve an equation. By the end of this guide you should be able to read the roots of an equation from where a curve crosses the x-axis, and solve a pair of equations from where two graphs intersect.

You should also be able to work out which line to draw to solve a given equation from an existing curve, estimate solutions to a sensible accuracy, explain why graphical answers are approximate, and use a graph to find a region or a turning point.

The organising idea is that a solution is a point where two graphs meet. Solving f(x) = 0 means finding where y = f(x) meets y = 0, which is the x-axis — that is why roots are read off there. Solving f(x) = 3 means finding where the same curve meets the horizontal line y = 3. And solving f(x) = g(x) means finding where the two curves meet each other. Once you see every equation as a question about an intersection, the hardest question type in this topic — "what line should be drawn to solve this equation?" — becomes a matter of rearrangement rather than insight.

Key terms and definitions

Root — a solution of an equation, where the expression equals zero.

Intersection — a point where two graphs cross.

f(x) — a function of x, whose graph is y = f(x).

Estimate — an approximate value, which is what a graph provides.

Turning point — a maximum or minimum on a curve.

Iteration — a numerical method used when a graph is not accurate enough.

Core concepts

Roots from a single graph

The solutions of f(x) = 0 are where the graph of y = f(x) crosses the x-axis, since that is where y is zero.

So the graph of y = x² − x − 6 crossing at x = −2 and x = 3 tells you those are the solutions of x² − x − 6 = 0.

The number of crossings gives the number of real solutions: a parabola may cross twice, touch once, or miss the axis entirely and have no real roots.

Reading the crossings is the whole method, and the accuracy depends on how carefully the curve was drawn.

Solving f(x) = a number

To solve an equation where the right-hand side is not zero, draw the horizontal line at that value and read where it meets the curve.

For x² − x − 6 = 4, draw y = 4 and read the x-coordinates of the two intersections.

That works because the equation asks where the curve's height equals 4, which is exactly what the horizontal line marks.

Marks are awarded for drawing the correct line as well as for the readings, so the line must appear on the graph.

Solving a pair of equations

Where two graphs are drawn on the same axes, their intersections give the solutions of the two equations solved together.

Two straight lines meet at one point, giving the solution of a pair of linear simultaneous equations.

A line and a curve may meet twice, giving two pairs of solutions, or touch once, or miss entirely.

Each intersection supplies both an x-value and a y-value, and questions usually want both, written as coordinates.

Working out which line to draw

This is the most demanding question type, and it always follows the same procedure.

You have a curve, say y = x² − 3x + 1, and are asked to solve a different equation, say x² − 4x + 3 = 0, using it.

Rearrange the equation you want to solve so that one side is exactly the expression already drawn.

Take x² − 4x + 3 = 0. Add x to both sides and subtract 2: x² − 3x + 1 = −x + 2.

The left side now matches the drawn curve, so the line to draw is y = −x + 2, and the x-coordinates where it meets the curve are the solutions.

Two checks make this reliable. The line you produce must be one you can actually draw, so it should be linear. And substituting a rough solution back into the original equation confirms the rearrangement was correct.

Why graphical answers are approximate

Every value read from a graph depends on the accuracy of the plotting, the smoothness of the curve and the precision of reading a scale by eye.

That is why such questions ask for answers to one decimal place rather than exact values, and why an answer quoted to three decimal places is claiming precision the method cannot deliver.

If a question wants exact answers, it expects an algebraic method — factorising, the formula, or completing the square — not a graph.

Where more accuracy is needed than a graph gives but algebra will not work, iteration is the intended method.

Reading to a sensible accuracy

Read to the nearest sensible division of the scale, and state the accuracy you are working to.

A grid with squares worth 0.2 supports readings to about one decimal place, not two.

Mark the readings on the graph with lines down to the x-axis. Those lines earn credit independently of the values, and they make the reading checkable.

Turning points and other features

A drawn graph also gives features that algebra would take longer to find.

The turning point is read directly as a pair of coordinates, giving the maximum or minimum value and where it occurs.

The y-intercept is read where the curve crosses the vertical axis.

The line of symmetry of a parabola runs vertically through the turning point, halfway between the roots.

In a context question these carry meaning: the maximum height of a projectile, the time it lands, the starting value.

Solving simultaneous equations graphically

A pair of linear simultaneous equations can be solved entirely from a graph, and some questions require it.

Draw both lines on the same axes, using a table of values or the gradient-and-intercept method for each, then read the coordinates of the crossing point.

For y = 2x + 1 and y = 5 − x, the lines meet at approximately (1.3, 3.7), and both coordinates form the answer.

The limitations are the usual ones. Parallel lines never cross, which shows graphically that the equations have no solution. And a crossing point falling between grid lines can only be estimated, which is why the algebraic method is preferred when exact values are wanted.

Using a graph to check algebra

Even where a question is solved algebraically, a sketch is a fast check on whether the answer is plausible.

An algebraic solution claiming a root at x = 7 for a curve that plainly crosses the axis near x = 2 signals an error before any marks are lost.

The same applies to the number of solutions: if factorising produced one root but the sketch crosses the axis twice, a solution has been dropped somewhere.

A rough sketch takes seconds and needs no table of values — the shape from the highest power, the y-intercept from the constant, and the approximate roots are enough.

Regions and inequalities

A graph can also answer an inequality rather than an equation.

The solution of f(x) < 0 is the range of x-values where the curve lies below the x-axis, read between the roots.

For f(x) > g(x), the solution is where the first graph sits above the second, read between their intersections.

Shade the region if the question asks for one, and give the answer as an inequality rather than a pair of values.

Worked examples

Example 1: Reading roots and solving a related equation

The graph of y = x² − 2x − 3 has been drawn. Use it to solve x² − 2x − 3 = 0 and then x² − 2x − 3 = 5.

For the first, read where the curve crosses the x-axis. It does so at x = −1 and x = 3, so those are the solutions.

For the second, draw the horizontal line y = 5 and read where it meets the curve. The intersections are at approximately x = −2 and x = 4.

Both readings should be marked with lines down to the axis, and the answers described as estimates.

Checking the second algebraically confirms it: x² − 2x − 8 = 0 factorises to (x − 4)(x + 2), giving exactly 4 and −2. ✓

Example 2: Finding the line to draw

The graph of y = x² + x − 2 is drawn. What line should be drawn to solve x² + 3x − 5 = 0?

Rearrange so the left side matches the drawn curve.

Starting from x² + 3x − 5 = 0, subtract 2x and add 3 to both sides: x² + x − 2 = −2x + 3.

The left-hand side is now the drawn curve, so the line to draw is y = −2x + 3.

The solutions are the x-coordinates where that line crosses the curve.

The check is that the required line is linear, which it is, and that the rearrangement is reversible — adding 2x and subtracting 3 returns the original equation.

Example 3: An inequality from a graph

The graph of y = x² − 4 is drawn, crossing the x-axis at −2 and 2. Use it to solve x² − 4 < 0.

The inequality asks where the curve lies below the x-axis.

Between the two roots, the curve dips below the axis; outside them it rises above.

So the solution is −2 < x < 2.

Note the answer is a range written as an inequality, not the two roots themselves — those are only the boundaries of the region.

Common mistakes and how to avoid them

Reading roots from the y-axis. Roots are where the curve crosses the x-axis.

Not drawing the required line. The line carries marks of its own.

Giving only the x-values for an intersection. A point of intersection has both coordinates.

Rearranging to a line that cannot be drawn. The required line must be linear.

Quoting a graphical answer too precisely. These are estimates, usually to one decimal place.

Giving the roots as the answer to an inequality. The answer is a range between or beyond them.

Using a graph when exact values are wanted. Switch to an algebraic method.

Exam technique for "Solving Equations Graphically"

Draw and label every line you use, and leave the reading lines down to the axis on the diagram. Both are worth marks independently of the values.

For "what line should be drawn" questions, write the rearrangement out line by line until the left side matches the printed curve exactly.

State the accuracy of your readings and describe them as estimates.

Give intersection answers as coordinate pairs when the question asks for solutions of two equations.

For inequalities, answer with a range and shade the region if asked.

Check a graphical answer algebraically when time allows — it costs little and confirms the reading.

Quick revision summary

A solution is a point where two graphs meet.

The roots of f(x) = 0 are where the curve crosses the x-axis. The number of crossings gives the number of real solutions.

To solve f(x) = a number, draw that horizontal line and read the intersections.

To solve two equations together, read where their graphs intersect, giving both coordinates.

To find which line to draw, rearrange the equation you want to solve until one side is exactly the expression already plotted; the other side is the line.

Graphical answers are estimates, normally to one decimal place. Exact answers need algebra, and greater numerical accuracy needs iteration.

For an inequality, find where the curve lies below or above the relevant line and give the answer as a range.