The $x$ and $y$ coordinates of points on a straight line have a numerical pattern. There are three numerical patterns to observe on this line:

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

The points are plotted such that the $x$ coordinates are going up in ones;
The $y$ coordinates are going up in $4$ s;

More importantly, there is a pattern between the two numbers in each ordered pair. That is, the $y$ coordinate is four times the $x$ coordinate then add one.

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

Also, $(5,21)$ . Notice, $4 \times 5 +1 = 21$ .

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

We can capture this calculation with an equation:

$y=4x+1$

Notice that the number we multiply $x$ by is 4. This is the same number the $y$ coordinates are going up in (when the $x$ coordinates go up in ones). We use the letter $m$ to represent this number (the multiplier); and we call it the slope because the multiplier determines how steep the line is.

As long as we know two points on a line, we can calculate the slope and then figure out the equation of the line.

To calculate slope, $m$ , we can use:

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

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

$y=mx + \dots$

If one of the points we know lies on the $y$ axis, that is, the point has $x=0$ , we call it the $y$ intercept of the line. This is the number that we complete the equation with.

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

We only need to use two points to find the equation. Let’s use $(0,7)$ because this is the $y$ intercept 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 $+7$

$y=3x+7$

We check 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:

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

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

We then substitute the $x$ and $y$ coordinates of one known point $(x_1,y_1)$ and the slope $m$ into the equation

$y_1=m\,x_1+b$

to calculate the value $b$ .

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

Solution: We need only two points to calculate the equation. Let’s use $(-1,20)$ 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

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A more efficient formula