The $x$ and $y$ coordinates of points on a straight line are related through a calculation.

Remember that the first number is the $x$ coordinate, the second is the $y$ coordinate, that is: $(x,y)$ .

Notice in the picture above:

The points are plotted such that the $x$ coordinates are going up in ones;
The $y$ coordinates are going up in $4$ s;
The $y$ coordinate is equal to the $x$ , times 4, plus 1.

Take the point $(2,9)$ . Notice, $2 \times 4 +1=9$ .

Take the point $(5,21)$ . Notice, $5 \times 4 +1 = 21$ .

This calculation works for all the coordinate pairs on the line.

We can capture this calculation with an equation $y = x \,\times 4 +1$ , which is written as:

$y=4x+1$

Calculating $y=mx+b$

To calculate an equation there are two steps:

Calculate $m$

We use the letter $m$ to represent the number that multiplies $x$ . We call it the slope because the multiplier determines how steep the line is.

In the example above, the $y$ -coordinates increase by $4$ , just like the four times table. That is the value $m$ .

We can also calculate $m$ with:

$m=\frac{\text{rise}}{\text{run}}$

or the formula

$m=\frac{y_2-y_1}{x_2-x_1}$

Once we know the slope, we know that the equation of the line begins

$y=mx + \dots$

2. Calculate the constant $b$

We complete the equation with a constant. There are different ways to calculate the constant. One way is to use the point that has $(0,?)$ . This is the $y$ -intercept. In the example above, the $y$ intercept is $(0,1)$ and so the equation above finishes with $+1$ .

Example: Determine the equation of the line that goes through the points $(0,7),(1,10),(2,13),(5,22),(10,37)$

Use any two points to calculate $m$ . Let’s use $(0,7)$ and $(10,37)$ .

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{37-7}{10-0}=\frac{30}{10}=3$

That tells us that $m=3$ and so our equation begins $y=3x + \dots$ .

The $y$ intercept has coordinates $(0,7)$ and so our equation finishes with $+7$

$y=3x+7$

Check the equation with some other coordinate pairs:

Use point $(3,16)$ : $y=3(3)+7=9+7=16 \checkmark$

Use point $(5,22)$ : $y=3(5)+7=15+7=22 \checkmark$ .

Practice:

In this practice applet, the point $(0,?)$ is provided, along with one other point.

GeoGebra applet link

Test out on Slope y intercept

Any Two Points

To find the equation of a line using any two points, we begin by calculating the slope with the formula

$m=\frac{y_2-y_1}{x_2-x_1}$

To calculate the constant $b$ , substitute the $x$ and $y$ coordinates of one known point $(x_1,y_1)$ and the slope $m$ into the equation

$y_1=m\,x_1+b$

Example: Find the equation of the line that goes through the points $(-1,20)$ , $(4,10)$ , and $(7,4)$ .

Use any two points: $(-1,20)$ and $(7,4)$ .

First, calculate slope:

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-20}{7-(-1)}=\frac{-16}{8}=-2$

Second, calculate $b$ using slope $m=-2$ , and any point. Let’s use $(7,4)$ .

$\begin{align*}y_1&=m\,x_1+b\\[10 pt]4&=-2(7)+b\\[10 pt]4&=-14+b\\[10 pt]18&=b\end{align}$

We now have $m=-2$ and $b=18$ , therefore the equation of the line is

$y=-2x+18$

which could also be written as

$y=18-2x$

just by changing the order of the terms.

Practice:

GeoGebra applet link

A more efficient formula

If we streamline the process of calculating $b$ , we arrive at the formula

$y-y_1=m(x-x_1)$

This is sometimes known as the ‘point/slope’ formula.

Using the same example as before, calculate the equation of the line that goes through the points $(-1,20)$ and $(7,4)$

We need to calculate the slope $m$ in the same way:

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-20}{7-(-1)}=\frac{-16}{8}=-2$

However, to calculate the equation we substitute $m=-2$ and choose any known point for the $(x_1,y_1)$ . Let’s use $(7,4)$ again:

$\begin{align*}y-y_1&=m(x-x_1)\\[10 pt]y-4&=-2(x-7)\\[10 pt]y-4&=-2x+14\\[10 pt]y&=-2x+18\end{align}$

Why it works

This formula $y-y_1=m(x-x_1)$ is derived as follows:

$\begin{align*}y_1&=m\,x_1+b\\[10pt]y_1-m\,x_1&=b\end{align}$

Now substitute $y_1-mx_1$ in place of $b$ in the formula $y=mx+b$ and rearrange:

$\begin{align*}y&=mx+b\\[10 pt]y&=mx+y_1-m\,x_1\\[10 pt]y-y_1&=mx-mx_1\\[10 pt]y-y_1&=m(x-x_1)\quad\text{as required.}\end{align}$

Use any method to calculate the equation given the coordinates of one point and the slope:

Printed practice:Grade 10 Point Slope

GeoGebra applet link

Interactive Math Resources for Classroom Discussion, Inquiry and Distance Learning

Linear Function Equation

Calculating $y=mx+b$

Calculate $m$

2. Calculate the constant $b$

Practice:

Any Two Points

A more efficient formula