Skip to main content

Section 14.1 Start Here, Go There

The slope-intercept form of a line is useful when using them in practical applications. However, when it comes to writing down the equation of a line, it is often inconvenient. The challenge is that for the slope-intercept form, the location of the initial point is restricted to being the and that's not always true of the available information.

We are going to consider the situation where we know one point on the line and we know what the slope of the line is. The known point will be labeled as \((x_0, y_0)\text{.}\) We are also going to pick another point on the line and call it \((x,y)\text{.}\) The goal is to find a mathematical relationship between these two points.

The only piece of information that has not been used is the slope. The slope is the same value no matter which two points on the line are chosen, so we will use these two points. The rise is the change in the \(y\)-coordinates and the run is the change in the \(x\)-coordinates. We calculate these by taking the differences between the corresponding coordinates.

We know that the slope is the change \(y\) in divided by the change in \(x\) so we can plug this in and then manipulate the formula.

\begin{equation*} m = \frac{y - y_0}{x - x_0} \implies y - y_0 = m(x - x_0) \end{equation*}

Definition 14.1.1. Point-Slope Form.

The point-slope form of the line that passes through the point \((x_0, y_0)\) with slope \(m\) is \(y - y_0 = m(x - x_0)\text{.}\)

Activity 14.1.1. Using the Point-Slope Form.

Once you have the formula, some problems involving the point-slope form of a line simply require to plug in values. For example, the point-slope form of the line that passes through the point \((1, -2)\) with slope \(\frac{4}{3}\) is \(y + 2 = \frac{4}{3} (x - 1)\text{.}\)

Try it!

Determine the point-slope form of the line that passes through the point \((-2, 5)\) with slope \(-2\text{.}\)

Solution.
\begin{equation*} y - 5 = -2(x + 2) \end{equation*}

Activity 14.1.2. Lines Have Multiple Representations in Point-Slope Form.

It turns out that a line has multiple representations using the point-slope form, depending on which point is considered to be the initial point. This is much easier to identify from a graph. Notice that the lines in the graphs below are identical.

Try it!

Find at least one more point-slope form for the line above.

Solution.
\begin{equation*} y - 3 = \frac{2}{3} (x - 4) \qquad \text{ or } \qquad y + 3 = \frac{2}{3} (x + 5) \end{equation*}

Activity 14.1.3. Rewriting Point-Slope Form in Slope-Intercept Form.

Because there are multiple representations of a single line, we say that the point-slope form of a line is not unique. This means that two different people may end up getting different answers to the same question, and they can both be correct. One way to check that the equations represent the same line is to rewrite the equations in slope-intercept form by solving for \(y\text{.}\)

Try it!

Write the equation \(y - 3 = \frac{2}{3} (x - 4)\) in slope-intercept form.

Solution.
\begin{equation*} \begin{aligned} y - 3 \amp = \frac{2}{3} (x - 4) \\ y - 3 \amp = \frac{2}{3} x - \frac{8}{3} \\ y \amp = \frac{2}{3} x + \frac{1}{3} \end{aligned} \end{equation*}