Interpret and use the equation y = mx + c, find the gradient and intercept, and determine the equation of a straight line

What this dot point is asking

SEAB wants you to understand the straight-line equation $y = mx + c$, find the gradient and the $y$-intercept, and determine a line's equation from a graph, from two points, or from a point and a gradient. Straight-line work underpins coordinate geometry and the reading of linear graphs.

The answer

The form y = mx + c

A straight line is described by:

where $m$ is the gradient (steepness) and $c$ is the $y$-intercept (the value of $y$ where the line crosses the $y$-axis). Reading these two numbers off the equation tells you everything about the line's position and slope.

Gradient

The gradient measures how steeply the line rises:

A positive gradient rises left to right, a negative gradient falls, and a zero gradient is horizontal. The steeper the line, the larger the magnitude of $m$.

Finding the equation

From two points, find the gradient first, then substitute one point into $y = mx + c$ to find $c$. From a graph, read the intercept directly off the $y$-axis and compute the gradient from any two clear grid points.

Parallel lines

Parallel lines have the same gradient. So any line parallel to $y = 3x + 2$ has gradient $3$ and differs only in its intercept $c$. This makes parallelism easy to test by comparing gradients.

Reading a real-life linear graph

In applied questions the gradient and intercept carry units and meaning, so interpreting them is as important as calculating them. On a distance-time graph the gradient is a speed (distance per unit time) and a horizontal section means stationary; on a cost-quantity graph the gradient is the price per item and the intercept is a fixed charge. So a line $C = 2n + 5$ for the cost of $n$ items says each item costs $2$ dollars and there is a fixed $5$-dollar charge. Translating the gradient and intercept back into the words of the problem, with their units, is exactly what an E-Maths interpretation question rewards.

Horizontal and vertical lines

Two special cases do not fit the usual $y = mx + c$ reading and are worth knowing. A horizontal line has gradient $0$ and equation $y = c$, because $y$ never changes. A vertical line has an undefined gradient (the run is zero, so the fraction is undefined) and equation $x = a$, because $x$ is constant while $y$ varies freely. Recognising that $y = 4$ is a flat line and $x = 4$ is an upright line prevents the common mix-up between the two, and explains why a vertical line cannot be written in the form $y = mx + c$ at all.

Examples in context

Example 1. Taxi fares. A fare of a fixed flag-down plus a rate per kilometre is a straight-line relationship, $\text{cost} = (\text{rate}) \times d + (\text{flag-down})$. The flag-down is the intercept and the per-kilometre rate is the gradient.

Example 2. Converting temperatures. The conversion from Celsius to Fahrenheit, $F = 1.8C + 32$, is a straight line with gradient $1.8$ and intercept $32$. Reading the gradient and intercept explains how the two scales relate.

Try this

Q1. State the gradient and $y$-intercept of $y = -4x + 9$. [2 marks]

• Cue. Gradient $-4$, intercept $9$.

Q2. Find the gradient of the line through $(0, 2)$ and $(5, 17)$. [2 marks]

• Cue. $m = \dfrac{17 - 2}{5 - 0} = 3$.

Q3. Write the equation of the line with gradient $2$ passing through $(0, -3)$. [2 marks]

• Cue. Intercept is $-3$, so $y = 2x - 3$.