GCSE · Maths · AQA · Spec 8300 · Foundation

Approximate solutions of equations from a graph

What if you could solve an equation without moving a single term? Draw it, and the answer is sitting on the graph, waiting for you to go and find it.

Where does 3x + 1 equal 9?

012340481216xy(2, 7)

x: 2. y: 7

Slide the point along the line. Stop when its y-value is as close to 9 as you can get, then read x.

The flat line y = 9: This is the right-hand side of the equation, drawn as its own line. Where it meets y = 3x + 1, the left side is worth exactly 9, so the x-value at that point solves 3x + 1 = 9.

Watch out: Try to land exactly on 9. You'll get close, just under or just over, but never spot on, because the true answer sits between 2.64 and 2.68. That's normal: a graph usually gives an approximate answer. Here x ≈ 2.7.

Which number is the answer?

Two lines cross. So what's the solution?

To solve 2x + 1 = 7 − x, each side is drawn as its own line: y = 2x + 1 and y = 7 − x. The two lines cross at the point (2, 5). The line y = 7 − x also crosses the x-axis at (7, 0).

What is the solution of 2x + 1 = 7 − x?
How sure are you?

x on both sides, with a bracket

Problem

Use a graph to solve 2(x + 3) = 10 − x.

Spot the slip

Where does this graph solution go wrong?

Use a graph to solve 3(x + 1) = 11 − x.

A student's working — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Turn each side into a line, find where they meet, and read off x.

What you need to know

  • Solving an equation with a graph means finding the x-value where the graph reaches the value you need.
  • With x on both sides, draw each side as its own line. The solution is the x-coordinate of the point where they cross.
  • Multiply out a bracket (or substitute into it carefully) before you plot that side.
  • Answers read from a graph are usually approximate, so check yours by substituting it into the original equation.

The big picture

Any equation asks one question: for which x are both sides equal? A graph answers it by showing where. Draw each side as its own line (a number on its own side becomes a flat line), find where the lines meet, and read the x-coordinate there. Deal with any bracket before you plot, and remember that a graph reading is usually an estimate, so check it by substituting back.

Key points

1An equation becomes a question about place: for which x do both sides give the same value?
2One side just a number, like 3x + 1 = 9? Draw y = 3x + 1 and the flat line y = 9.
3x on both sides? Draw y = (left side) and y = (right side) on the same axes.
4The answer is the x-coordinate of the crossing: not its y-coordinate, and not where a line meets the x-axis.
5Brackets: the number outside multiplies every term inside, so 2(x + 3) is 2x + 6.
6Graph readings are usually estimates. In the original equation, your x should make the two sides come out equal or nearly equal.

Worked example

Problem

Use a graph to solve 3(x − 2) = 5.

⚠ Watch out

Giving the y-coordinate of the crossing point as the answer. The y-value is what both sides are equal to; the solution is the x-value that makes them equal.

🧠

Memory hook

Where the lines meet, drop down to the street: the x-axis holds your answer.

✓

Check yourself

Could you solve 4x − 1 = 2(x + 3) with a graph? Name the two lines, say which coordinate of the crossing gives the answer, and how you'd check it.

Flashcards

(8)
What are you looking for when you solve an equation with a graph?
The x-value where the graph reaches the value you need, or where the lines for the two sides meet.
To solve 2x − 1 = 6 using the graph of y = 2x − 1, which line do you add?
The flat line y = 6. Read the x-value where it crosses y = 2x − 1: x = 3.5.
Which two lines would you draw to solve 5x − 2 = 3x + 4?
y = 5x − 2 and y = 3x + 4, on the same axes. The x-coordinate where they cross is the solution (x = 3).
The lines for the two sides of an equation cross at (4, −1). What is the solution?
x = 4. The −1 is the value both sides equal there, not the answer.
The line y = 8 − x crosses the x-axis at x = 8. Which equation does that solve?
8 − x = 0. That's a different equation from one like 8 − x = x + 2, so the x-axis crossing isn't its answer.
How do you plot y = 4(x − 1)?
Multiply out first: y = 4x − 4, because the 4 multiplies both terms. Or substitute into the bracket: at x = 3, 4(3 − 1) = 8.
Why is a solution read from a graph usually approximate?
The crossing point often falls between gridlines, so you can only estimate it, for example to 1 decimal place.
How do you check a solution you read from a graph?
Substitute it into both sides of the original equation. If they come out nearly equal, your reading is good.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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How this lesson was checked. This AQA GCSE Maths (specification 8300)lesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 29 September 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.