The Gradient Intercept Equation
When a line is written as \(y = mx + c\) (gradient-intercept form), the number \(m\) is the gradient and \(c\) is the \(y\)-intercept, so the equation shows at a glance how steep the line is and where it crosses the \(y\)-axis. This NSW Year 9 Core maths subtopic is Stage 5 Core content under NESA outcome MA5‑LIN‑C‑02.
You will read \(m\) and \(c\) from equations such as \(y = -x - 4\) and \(y = 5 - 2x\), build an equation from a gradient and an intercept, and find the equation of a graphed line. You will rearrange forms such as \(2x + 3y = 6\) and \(\frac{x}{3} + \frac{y}{5} = 1\) into \(y = mx + c\), compare the steepness of two lines, find a line parallel to another, and use a plumber's charge of a call-out fee plus an hourly rate. It is part of Year 9 Core Straight Line.
Every question with a fully worked solution.
- slope-intercept equation from graph Watch
Theory
The gradient-intercept form of a straight line is \(y = mx + c\), where \(m\) is the gradient and \(c\) is the \(y\)-intercept. This Year 9 maths page shows how to read \(m\) and \(c\) from an equation, write the equation of a line, rearrange other forms into \(y = mx + c\), and graph a line quickly.
Any straight line that is not vertical can be written as \(y = mx + c\). This is the gradient-intercept form.
\(m\) is the gradient: the number multiplying \(x\), including its sign. \(c\) is the \(y\)-intercept: the value of \(y\) when \(x = 0\), where the line crosses the \(y\)-axis.
If no number is written in front of \(x\), the gradient is \(1\) (or \(-1\) for \(-x\)). If there is no number on its own, \(c = 0\) and the line passes through the origin.
Lines with the same gradient are parallel. The larger the gradient (ignoring its sign), the steeper the line.
How to work with y = mx + c
- Rearrange the equation so that \(y\) is on its own, if it is not already.
- Read \(m\), the number multiplying \(x\) with its sign, and \(c\), the number on its own.
- To graph: mark \((0, c)\), then step out the gradient as rise over run to a second point. Rule the line.
- To build an equation: substitute the gradient and the \(y\)-intercept into \(y = mx + c\). A parallel line keeps the same \(m\).
- Check that the sign of \(m\) matches the direction of the line.
(i) \(y = -3x + 8\)
(ii) \(y = 7 - x\).
The gradient is the number multiplying \(x\), sign included. The \(y\)-intercept is the number on its own.
| \(\text{(i)}\ m\) | \(=\) | \(\mathbf{-3}, \quad c = \mathbf{8}\) |
(ii) Write it the usual way round first: \(y = -x + 7\). The \(x\) has no number in front, so the coefficient is \(-1\).
| \(\text{(ii)}\ m\) | \(=\) | \(\mathbf{-1}, \quad c = \mathbf{7}\) |
Both lines have negative gradients, so both fall from left to right.
Parallel lines have the same gradient, so \(m = \dfrac{2}{3}\). The \(y\)-intercept gives \(c = -4\). Substitute both into \(y = mx + c\).
| \(y\) | \(=\) | \(mx + c\) |
| \(y\) | \(=\) | \(\mathbf{\dfrac{2}{3}x - 4}\) |
Get the \(y\)-term alone, then divide every term by the number multiplying \(y\).
| \(3x - 4y\) | \(=\) | \(8\) |
| \(-4y\) | \(=\) | \(-3x + 8\) |
| \(y\) | \(=\) | \(\dfrac{3}{4}x - 2\) |
The gradient is \(\mathbf{\dfrac{3}{4}}\) and the \(y\)-intercept is \(\mathbf{-2}\). Dividing \(-3x\) and \(8\) by \(-4\) flips both signs.
Mark the \(y\)-intercept, then step out the gradient as rise over run.
| \(c\) | \(=\) | \(5 \;\Rightarrow\; (0, 5)\) |
| \(m\) | \(=\) | \(-2 = \dfrac{-2}{1}\) |
From \((0, 5)\) go \(1\) across and \(2\) down to \((1, 3)\). Rule the line through the two points.
Common pitfalls
Frequently asked questions
What does y = mx + c mean?
It is the gradient-intercept form of a straight line. \(m\) is the gradient, the steepness of the line, and \(c\) is the \(y\)-intercept, where the line crosses the \(y\)-axis.
How do you find the gradient from an equation?
Rearrange the equation into \(y = mx + c\). The gradient is the number multiplying \(x\), including its sign. For \(y = -3x + 8\) the gradient is \(-3\).
How do you write the equation of a line from the gradient and y-intercept?
Substitute them into \(y = mx + c\). A line with gradient \(\dfrac{2}{3}\) and \(y\)-intercept \(-4\) is \(y = \dfrac{2}{3}x - 4\).
How do you graph a line using the gradient and y-intercept?
Plot the \(y\)-intercept \((0, c)\). Write the gradient as rise over run and step it out from that point to get a second point. Rule a straight line through both.
Do parallel lines have the same gradient?
Yes. Parallel lines have equal gradients. They have different \(y\)-intercepts, otherwise they would be the same line.