The and 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 , the second is for , that is: .

- The points are plotted such that the coordinates are going up in ones;
- The coordinates are going up in s;

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

Take the point . Notice, .

Also, . Notice, .

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

We can capture this calculation with an equation:

Notice that the number we multiply by is 4. This is the same number the coordinates are going up in (when the coordinates go up in ones). We use the letter 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, , we can use:

or the formula

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

If one of the points we know lies on the axis, that is, the point has , we call it the 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

We only need to use two points to find the equation. Let’s use because this is the intercept and .

That tells us that and so our equation begins .

The intercept has coordinates and so our equation finishes with

We check with some other coordinate pairs:

Use point :

Use point : .

Practice:

# Any Two Points

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

We then substitute the and coordinates of one known point and the slope into the equation

to calculate the value .

**Example:** Find the equation of the line that goes through the points , , and .

Solution: We need only two points to calculate the equation. Let’s use and .

First, calculate slope:

Second, calculate using slope , and any point. Let’s use .

We now have and , therefore the equation of the line is

which could also be written as

just by changing the order of the terms.

**Practice:**

# A more efficient formula

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

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 and

We need to calculate the slope in the same way:

However, to calculate the equation we substitute and choose any known point for the . Let’s use again:

**Why it works**

This formula is derived as follows:

Now substitute in place of in the formula and rearrange:

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

Printed practice:Grade 10 Point Slope