Learning Objectives

Chapter Overview

Key skills you will develop by the end of Chapter 4.

Find the gradient of a straight line using $\frac{\text{rise}}{\text{run}}$

State the $y$-intercept $c$ from $y = mx + c$

Draw graphs of linear equations in the form $ax + by = k$

Use distance–time graphs, where the gradient is speed

Solve simultaneous equations by the graphical method

Solve simultaneous equations by elimination and substitution

Mathematics: Graphs of Linear Equations & Simultaneous Equations

Complete chapter notes with gradient, all methods, worked examples and full step-by-step exercises: PDF format

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Please note: This page is a concise preview with live interactive graphs. The complete notes with all worked examples, exercises and step-by-step solutions are in the PDF. Download it for the full learning material.

Interactive Graphs

Play With Live Graphs

Move the sliders — the graph redraws instantly. See exactly how each number in the equation changes the line.

These graphs are fully dynamic: as you change the values in an equation, the line (and the answer) updates live on the graph paper — just like drawing it yourself.

1. Gradient & y-intercept explorer: y = mx + c

Change the gradient $m$ and the y-intercept $c$, and watch the straight line move. The orange triangle shows the rise and the run.

Scale: 1 small box = 1 unit on each axis

Gradient m 2

y-intercept c 1

Equation: y = 2x + 1 · gradient = 2, cuts the y-axis at 1

2. Line plotter: ax + by = k

Set $a$, $b$ and $k$. The line is drawn and its x-intercept and y-intercept are marked automatically.

Scale: 1 small box = 1 unit on each axis

a 2

b -3

k 2

Line: 2x − 3y = 2
x-intercept (1, 0) · y-intercept (0, −0.67)

3. Graphical method: solve two equations at once

Two lines are drawn together. The green dot is where they cross — that point is the solution $(x, y)$. If the lines are parallel there is no solution; if they lie on top of each other there are infinitely many.

Line 1 Line 2

Scale: 1 small box = 1 unit on each axis

Line 1: a₁ 1

b₁ 1

c₁ 3

Line 2: a₂ 3

b₂ 1

c₂ 5

x + y = 3 and 3x + y = 5
Solution: x = 1, y = 2

Part 1 · Gradient & Graphs

Gradient & Drawing Lines

The straight-line basics: gradient, y-intercept, and drawing any line.

1. What do $m$ and $c$ mean in $y = mx + c$?

• Define gradient and y-intercept• What is m? What is c?

In the equation of a straight line $y = mx + c$: $m$ is the gradient (how steep the line is) and $c$ is the y-intercept (where the line cuts the y-axis).

gradient $m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$

📈 A line sloping up left→right has a positive gradient; sloping down has a negative gradient.

2. How do you find the gradient from two points?

• Gradient between two coordinates• rise over run

Pick any two points on the line and use $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.

Example: for $A(-2, 5)$ and $B(4, -7)$: $m = \dfrac{-7 - 5}{4 - (-2)} = \dfrac{-12}{6} = -2$.

⛔ Common Mistake: keep the order the same on top and bottom — $(y_2 - y_1)$ over $(x_2 - x_1)$, not mixed up.

3. How do you draw the graph of $ax + by = k$?

• Table of values method• How many points to plot?

Make a small table of values: choose a few $x$ values, work out each $y$, plot the points and join them with a ruler.

Example: $2x + y = 6$ gives $(0, 6)$, $(1, 4)$, $(3, 0)$.

📝 A straight line needs only 2 points, but always plot 3 — the third checks for mistakes. Try it live in the “ax + by = k” graph above.

4. On a distance–time graph, what does the gradient tell you?

• What is the gradient of a distance-time graph?• How to read speed

On a distance–time graph the gradient is the speed. A steep line means fast; a flat (horizontal) line means at rest (speed 0); a line sloping down means returning.

speed $= \dfrac{\text{distance}}{\text{time}}$

Part 2 · Simultaneous Equations

Three Ways to Solve a Pair

Graphical, elimination and substitution — the same answer, three routes.

5. What is the graphical method for simultaneous equations?

• How do graphs give the solution?• Point of intersection

Draw both lines on the same axes. The point where they cross satisfies both equations, so its coordinates $(x, y)$ are the solution.

Try the “solve two equations” graph above — the green dot is the solution and it moves as you change the numbers.

📝 Parallel lines never meet → no solution. The same line (one on top of the other) → infinitely many solutions.

6. How does the elimination method work?

• Add or subtract equations• Steps for elimination

Make the coefficient of one variable equal in size in both equations (multiply if needed), then add or subtract so that variable cancels.

Example: $3x - y = 12$ and $2x + y = 13$. Adding: $5x = 25$, so $x = 5$; then $y = 3$.

⛔ Common Mistake: if the signs of the matched variable are the same, subtract; if opposite, add.

7. How does the substitution method work?

• Make one variable the subject• Steps for substitution

Step 1: make one variable the subject of one equation.
Step 2: substitute it into the other equation and solve.
Step 3: substitute back to find the second variable.

Example: $7x - 2y = 21$ and $4x + y = 57$. From the second, $y = 57 - 4x$; substituting gives $15x = 135$, so $x = 9$, $y = 21$.

8. How do you solve a word problem with two unknowns?

• Forming equations from a story• Real-life simultaneous equations

Step 1: let two letters stand for the two unknowns.
Step 2: form two equations from the facts.
Step 3: solve them, then answer in words.

Example: a belt and a wallet cost \$42; 7 belts and 4 wallets cost \$213. Then $b + w = 42$ and $7b + 4w = 213$, giving a belt \$15 and a wallet \$27.

Vocabulary

Key Terms

Quick reference for the important words in this chapter.

Term	Meaning
Gradient $(m)$	The steepness of a line, $\dfrac{\text{rise}}{\text{run}}$.
y-intercept $(c)$	Where the line cuts the y-axis (the $y$ value when $x = 0$).
Linear equation	An equation whose graph is a straight line, e.g. $ax + by = k$.
Simultaneous equations	Two equations solved together for one pair $(x, y)$.
Point of intersection	Where two lines cross — the graphical solution.
Elimination	Adding/subtracting equations to remove one variable.
Substitution	Putting one variable in terms of the other, then solving.

Chapter Summary

Methods at a Glance

The three ways to solve a pair of simultaneous equations.

Method	How it works & when to use it
Graphical	Draw both lines; read the point of intersection. Good for seeing the answer; less exact for fractions.
Elimination	Match a variable’s coefficient, then add/subtract to cancel it. Best when coefficients match easily.
Substitution	Make one variable the subject, substitute, solve. Best when one variable already has coefficient $1$.
No solution	Lines are parallel (same gradient, different intercept).
Infinitely many	Both equations give the same line.

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Term	Meaning
Gradient \((m)\)	The steepness of a line, \(\dfrac{\text{rise}}{\text{run}}\).
y-intercept \((c)\)	Where the line cuts the y-axis (the \(y\) value when \(x = 0\)).
Linear equation	An equation whose graph is a straight line, e.g. \(ax + by = k\).
Simultaneous equations	Two equations solved together for one pair \((x, y)\).
Point of intersection	Where two lines cross — the graphical solution.
Elimination	Adding/subtracting equations to remove one variable.
Substitution	Putting one variable in terms of the other, then solving.