Equations of linear graphs

What y = mx + c means

Every straight line that is not vertical has an equation that can be written

$$y=mx+c$$

The equation is a rule that every point on the line obeys: put in the $x$-coordinate of any point on the line and it gives that point's $y$-coordinate.

• $m$ is the gradient: how much $y$ changes for every 1 unit increase in $x$.
• $c$ is the $y$-intercept: where the line crosses the $y$-axis, at the point $(0,c)$.

Why the two numbers mean this. Put $x=0$ into $y=mx+c$ and you get $y=c$, so the line passes through $(0,c)$, which is on the $y$-axis. Now increase $x$ by 1: $mx$ goes up by exactly $m$, so $y$ goes up by $m$. That is the definition of the gradient.

For example, $y=6x+3$ has gradient $6$ and $y$-intercept $3$: it crosses the $y$-axis at $(0,3)$ and rises 6 for every 1 across.

Reading the gradient and intercept from an equation

Before you read anything off, make sure the equation starts "$y=$", with $y$ on its own. Then:

• the gradient is the number multiplying $x$, including its sign;
• the $y$-intercept is the number on its own, including its sign.

The terms can come in either order. $y=7-4x$ is the same line as $y=-4x+7$, so its gradient is $-4$ and its $y$-intercept is $7$. A common slip is to take the first number you see as the gradient.

Watch the hidden numbers:

• $y=x-5$: the gradient is $1$, because $x$ means $1x$.
• $y=-x$: the gradient is $-1$ and the $y$-intercept is $0$. The line goes through the origin.
• $y=\frac{1}{2}x+4$: the gradient is $\frac{1}{2}$.

Vertical and horizontal lines

Two kinds of line have very short equations.

• Horizontal lines have equations $y=k$, for a number $k$. Every point on the line $y=3$ has $y$-coordinate 3, wherever it is across. Its gradient is $0$: it is $y=0x+3$.
• Vertical lines have equations $x=k$. Every point on the line $x=3$ has $x$-coordinate 3. A vertical line cannot be written as $y=mx+c$, because its gradient cannot be calculated.

The trap is the letter. The line $x=3$ is vertical, even though it "contains" an $x$: it is the set of points whose $x$-coordinate is 3, and they sit one above another. The $x$-axis itself is the line $y=0$, and the $y$-axis is the line $x=0$.

Finding the equation from a graph

To write down the equation of a straight line drawn on a grid:

1. Find the gradient $m$ from two clear grid points (rise over run, using the axis scales).
2. Read the $y$-intercept $c$: the $y$-value where the line crosses the $y$-axis.
3. Write $y=mx+c$ with your two numbers in it.
4. Check with another point on the line: its coordinates should fit the equation.

If the line crosses the $y$-axis off the visible grid, or not at a clear point, use one point and the gradient to work back to $x=0$. A question asking for "the equation of the line" wants a full equation, starting "$y=$", not just the two numbers.

Interpreting the gradient and intercept

Many straight-line graphs describe something real, and the two numbers then have meanings.

• The gradient is a rate: the change in the $y$ quantity for each one unit of the $x$ quantity. Its units are "$y$ units per $x$ unit".
• The intercept is the starting value: the value of $y$ when $x=0$.

For example, if a plumber charges $C=45h+60$ dollars for $h$ hours of work, the gradient 45 means 45 dollars for each hour, and the intercept 60 means a fixed call-out charge of 60 dollars, paid even for no time at all.

A negative gradient means the quantity is falling. If the fuel in a tank is $F=50-4t$ litres after $t$ hours, the tank starts with 50 litres and loses 4 litres every hour. It is empty when $F=0$: $50-4t=0$, so $t=12.5$ hours.

When you explain a gradient in words, always say what changes, by how much, and for each what: "the cost goes up by 45 dollars for each extra hour".

Fully simplified answers

Cambridge expects the equation of a line in a fully simplified form. That means:

• $y$ on its own on the left: $y=2x+3$, not $y-3=2x$;
• like terms collected: $y=2x+3$, not $y=2x+1+2$;
• no common factor left over: $2y=4x+6$ simplifies, by dividing every term by 2, to $y=2x+3$;
• no brackets left unexpanded: $y=2(x+1)+1$ is not finished.

A fraction gradient is fine: $y=\frac{2}{3}x-1$ is fully simplified. So is $y=5-x$; the order of the terms does not matter.