Solving Equations by Graphical Methods

Solving equations from a drawn curve: reading the x-axis, rearranging to find the straight line to draw, for quadratics and cubics.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1, Edexcel IGCSE Maths 4MA1 · 72 real past-paper questions below

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Where the Graphs Cross

At a crossing point, both equations are true at once. So the x values there solve the equation.

Rule

To solve with a graph you already have, make one side the curve's equation. The other side is the line to draw.

Key idea

If the other side is 0, the line is the x-axis. Read where the curve crosses it.

The Curve on Its Own

The simplest questions: the equation is the curve's own equation, set equal to 0 or to a number. No algebra needed — just read the graph.

Worked example

1
$y = x^{2} - 2 x - 4$ — The curve already drawn on the grid. We use it to solve equations by reading, not by algebra.
2
$x^{2} - 2 x - 4 = 0$ — The equation is the curve's equation with y replaced by 0. The line y = 0 is the x-axis.
3
$x \approx - 1.2 or x \approx 3.2$ — Read down from the two places the curve crosses the x-axis. Two crossings, so two answers.

Now the curve equals a number.

Worked example

1
$x^{2} - 2 x - 4 = 2$ — Same curve, but now it equals 2, not 0. So the line to draw is y = 2.
2
$y = 2$ — Rule a flat line across the grid at height 2. Every point on it has y = 2.
3
$x \approx - 1.6 or x \approx 3.6$ — Read down from both crossings to the x-axis. The answers are these x values, not the 2.

Key idea

If the equation is $x^{2} - 2 x = 5$ , take 4 from both sides first: $x^{2} - 2 x - 4 = 1$ . Now it is the curve equal to a number, so draw y = 1.

Reciprocal and Other Curves

Exams also use curves like $y = \frac{6}{x}$ . The method does not change: the crossing points give the answers.

Worked example

1
$y = \frac{6}{x}$ — The curve on the grid. It has two separate branches, because you cannot divide by 0.
2
$\frac{6}{x} = x + 1$ — The equation to solve. The left side is already the curve, so the right side is the line.
3
$y = x + 1$ — Rule this line using two points: x = 0 gives y = 1, and x = 2 gives y = 3.
4
$x = - 3 or x = 2$ — The line crosses each branch once. Read down from both crossings.
5
$\frac{6}{2} = 3 and 2 + 1 = 3$ — Check x = 2: both sides give 3, so it really solves the equation.

Common mistake

Stopping after one crossing. Look at both branches — the second answer is often on the other side of the y-axis.

Key idea

Any curve works the same way. To solve $2^{x} = 6$ from the graph of $y = 2^{x}$ , draw y = 6 and read x ≈ 2.6.

Finding the Line (Extended/Higher)

Most marks come from questions like “by drawing a suitable straight line, solve …”. You change the equation until one side is exactly the curve.

Worked example

1
$y = x^{2} - 3 x - 2$ — The curve on the grid. Our goal: make $x^{2} - 3 x - 2$ appear on one side of the new equation.
2
$x^{2} - 4 x + 1 = 0$ — The equation to solve. It is not the curve yet: it has −4x and +1, not −3x and −2.
3
$x^{2} - 3 x + 1 = x$ — Add x to both sides. −4x + x = −3x, which matches the curve.
4
$x^{2} - 3 x - 2 = x - 3$ — Take 3 from both sides. +1 − 3 = −2, so the left side is now exactly the curve.
5
$y = x - 3$ — The left side is y, so the right side is the line to draw.
6
$x = 0 \Rightarrow y = - 3, x = 4 \Rightarrow y = 1$ — Find two points on the line, then rule it right across the grid.
7
$x \approx 0.3 or x \approx 3.7$ — Read down from where the line crosses the curve. These x values solve the equation.

Common mistake

Giving the y values. The solutions are the x values of the crossing points.

Key idea

Write the line's equation, y = x − 3, in your answer. It earns a mark for working even if your readings are a little off.

Key idea

Backwards questions: if $y = x^{2} - 3 x - 2$ and $y = x - 3$ cross, their x values solve $x^{2} - 3 x - 2 = x - 3$ , which tidies to $x^{2} - 4 x + 1 = 0$ .

Multiply or Divide First (Extended/Higher)

Sometimes the equation has a different number in front of $x^{2}$ or $x^{3}$ . Adding cannot fix that, so multiply or divide first.

Worked example

1
$y = 2 x^{2} + 2 x - 3$ — The curve on the grid. Notice it starts with 2x².
2
$x^{2} - 2 x - 1 = 0$ — The equation starts with just x². Adding or taking away can never turn x² into 2x².
3
$2 x^{2} - 4 x - 2 = 0$ — Multiply every term by 2 (0 × 2 is still 0). Now it starts with 2x², like the curve.
4
$2 x^{2} + 2 x - 2 = 6 x$ — Add 6x to both sides: −4x + 6x = 2x, which matches the curve.
5
$2 x^{2} + 2 x - 3 = 6 x - 1$ — Take 1 from both sides: −2 − 1 = −3. The left side is now exactly the curve.
6
$y = 6 x - 1$ — So this is the line to draw. Points: x = 0 gives −1, x = 1 gives 5.
7
$x \approx - 0.4 or x \approx 2.4$ — Read down from where the line crosses the curve.

Common mistake

Multiplying only the x² term. Every term on both sides gets multiplied.

Key idea

If the equation has double the curve — say $2 x^{3}$ when the curve has $x^{3}$ — divide every term by 2 instead.

A Cubic Curve (Extended/Higher)

The 4-mark version often uses a cubic. The method is the same, but a cubic can cross a line three times.

Worked example

1
$y = x^{3} - 4 x + 1$ — The curve on the grid. Our goal: make $x^{3} - 4 x + 1$ one side of the equation.
2
$x^{3} - 5 x + 3 = 0$ — The equation to solve. It has −5x and +3, but the curve has −4x and +1.
3
$x^{3} - 4 x + 3 = x$ — Add x to both sides. −5x + x = −4x, which matches the curve.
4
$x^{3} - 4 x + 1 = x - 2$ — Take 2 from both sides. +3 − 2 = +1, so the left side is now the curve.
5
$y = x - 2$ — The right side is the line to draw. Points: x = 0 gives −2, x = 3 gives 1.
6
$x \approx - 2.5, 0.7, 1.8$ — The line cuts the cubic three times. Give all three x values.

Key idea

Check your line at x = 0: the curve gives 1 and the line gives −2. Curve minus line is 3, the equation's value at x = 0. If that fails, your line is wrong.

How Many Solutions? (Extended/Higher)

Some questions ask how many solutions an equation has, or for which k it has none. Count how often a flat line y = k meets the curve.

Worked example

1
$y = x^{2} - 2 x - 4$ — The curve from earlier. Its lowest point is (1, −5).
2
$x^{2} - 2 x - 4 = k$ — The curve equals a number k, so its solutions are where the flat line y = k meets the curve.
3
$k > - 5 : two solutions$ — A flat line above the lowest point crosses the curve twice.
4
$k = - 5 : one solution$ — A line through the lowest point only touches the curve, at x = 1.
5
$k < - 5 : no solutions$ — A line below the lowest point misses the curve. So k = −6 is an integer with no solutions.

Key idea

For a cubic like $y = x^{3} - 4 x + 1$ , the turning points are at about y = 4.1 and y = −2.1. A flat line between them crosses 3 times. Exactly 2 solutions means the line goes through a turning point, so k ≈ 4.1 or k ≈ −2.1.

Which Method? Read the Question

Compare the equation with the curve on the grid.

You see	You do
Curve = 0	read the x-axis crossings
Curve equals a number	draw a flat line at that number
A different equation	rearrange: curve on one side
A different number in front of x² or x³	multiply or divide first
How many solutions?	count where a flat line y = k meets the curve
Two lines crossing	Graphical Solution of Simultaneous Equations

Key idea

Give answers to 1 decimal place, give every crossing, and write down the line you drew.

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