Writing Linear Equations in Point-Slope Form

Objective Learn how to write equations for lines given information such as the slope and a point on the line, or two points on the line.

There are a number of ways to write equations representing straight lines. This lesson introduces two of them. The point-slope form can be written if the slope of the line in question and a point on the line are given, or if two points on the line are given. The standard form is also useful because it can be used to represent any line, even a vertical one. Vertical lines cannot be given in point-slope form since their slope is not defined.

Point-Slope Form

Key Idea

If we have a straight line, and we choose any two different points on it, we will get the same value for the slope no matter what points we choose.

Now draw a line like the one shown on the figuer below.

Display some points on the line, such as (1, 2) and (2, 4), in addition to the origin (0, 0). Then, try to find the slope of this line (it is actually 2). We write the general point P = ( x , y ), where x and y are variables. "How can you tell from the coordinates x and y whether P lies on the line?"

Remember that if (0, 0) and ( x , y ) are used to compute the slope, the answer still must be 2, since the key idea states that we will get the same answer no matter which two points are chosen.

You can find the slope by dividing the rise by the run. Since the rise is y and the run is x , slope

We know that the slope of this line is 2, so .

So, the rise is twice the run, and y = 2x for any point on this line. This equation holds for the coordinates of any point on the line.

There are many equations that describe the same line. We could have computed the slope using (x, y) and any other point on the line, such as (1, 2), as shown below.

(y - 2) = 2( x - 1)	Multiply each side by x - 1.

Try to write your own equations for the line, using various points, such as (2, 4) or (3, 6).

Here are some more examples. Determine the slope and write an equation for each line graphed.

Example 1

Write an equation for the line.

Solution

The line has slope 3. One equation for it is y = 3x .

Example 2

Write an equation for the line.

Solution

The line has slope and an equation for it is .

So, if the slope of a line (let's call it m) and a single point (let's call it ( x ₀ , y ₀ )) on that line are known, then the following equation for the line can be written.

( y - y ₀ ) = m ( x - x ₀ )

This equation is said to be in point-slope form since it uses only information about one point and the slope.

Now, work on the following excercises.

Exercises

Find an equation in point-slope form for each line.

1. the line with slope 4 going through the point (1, 2)

( y - 2) = 4( x - 1)

2. the line with slope  1 going through the point (5, 4)

( y - 4) = -1( x - 5)

3. the line with slope going through the point (0, 10)

The slope and a single point on the line determine the whole line. Intuitively, the steepness and one point determine the line. This can be illustrated by rotating a ruler through a single point, and tracing out many of the possible slopes.