Linear Equations

What you will learn

• Read gradient and intercept from y = mx + c.
• Draw a straight line from an equation.
• Find a line through two points and solve simultaneous equations.

Before you start: you should know coordinates and how to rearrange formulas. See Coordinates in Four Quadrants.

The idea

An equation is linear when the variables appear only to the first power: no $x^2$, no $\sqrt{x}$, no $\frac{1}{x}$. Plot all the points $(x, y)$ that satisfy a linear equation in two variables and you always get a straight line, which is where the name comes from.

Gradient and intercept

Every non-vertical straight line can be written as

$y = mx + c$

• $m$ is the gradient (slope): how much $y$ goes up for each step of $1$ to the right. A positive gradient rises, a negative gradient falls, and a gradient of $0$ is flat.
• $c$ is the y-intercept: where the line crosses the $y$-axis, because when $x = 0$, $y = c$.

The line $y = 2x - 3$ starts at $-3$ on the $y$-axis and climbs $2$ units for every $1$ unit across.

Gradient as “rise over run”

Between any two points on a line, the gradient is the change in $y$ divided by the change in $x$. It doesn’t matter which two points you pick. That constant rate of change is what makes the line straight.

Worked examples

Example 1: Line through two points

Find the equation of the line through $(1, 5)$ and $(4, 14)$.

1. Gradient: $m = \dfrac{14 - 5}{4 - 1} = \dfrac{9}{3} = 3$.
2. So $y = 3x + c$. Substitute the point $(1, 5)$: $5 = 3(1) + c$.
3. Solve: $c = 2$.
4. Check with the other point: $3(4) + 2 = 14$. ✓

Answer: $y = 3x + 2$

Example 2: Simultaneous equations

Solve $y = 2x + 1$ and $x + y = 10$ together.

1. Both equations describe lines. The solution is the point where they cross.
2. Substitute the first into the second: $x + (2x + 1) = 10$.
3. Simplify and solve: $3x + 1 = 10$, so $3x = 9$ and $x = 3$.
4. Find $y$: $y = 2(3) + 1 = 7$.
5. Check in the second equation: $3 + 7 = 10$. ✓

Answer: $x = 3$, $y = 7$. The lines meet at $(3, 7)$.

Example 3: Reading a real graph

A plumber charges a £40 call-out fee plus £30 per hour. Write the cost $C$ for $h$ hours as a linear equation.

1. The fixed amount is the intercept: $c = 40$.
2. The rate per hour is the gradient: $m = 30$.

Answer: $C = 30h + 40$. A 2.5-hour job costs $30(2.5) + 40 = £115$.

Vertical lines like $x = 4$ have an undefined gradient and cannot be written as $y = mx + c$.

Why it matters

Any situation with a fixed starting amount and a constant rate is linear: taxi fares, mobile tariffs, a car travelling at steady speed, converting Celsius to Fahrenheit ($F = 1.8C + 32$). Businesses use simultaneous linear equations to find break-even points, and scientists draw lines of best fit through data to spot trends.

Linear equations are also the first example of a function, and the gradient is the first step towards the derivative in calculus.