# 5.2: Linear Equations in Point-Slope Form

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Write an equation in point-slope form.
- Graph an equation in point-slope form.
- Write a linear function in point-slope form.
- Solve real-world problems using linear models in point-slope form.

## Vocabulary

Terms introduced in this lesson:

- point-slope form

## Teaching Strategies and Tips

Students learn to write linear equations in point-slope form given:

- The slope and any one point on the line (possibly the \begin{align*}y-\end{align*}intercept). See Examples 1, 2, and 8.
- Any two points on the line (\begin{align*}m\end{align*} is not given). See Examples 3 and 7.

An equation in point-slope form:

- Uses subscripts on \begin{align*}x\end{align*} and \begin{align*}y\end{align*} to designate the fixed, given point. \begin{align*}x\end{align*} and \begin{align*}y\end{align*} assume any other points on the line.
- Is not solved for \begin{align*}y\end{align*}. Suggest that students generate other values of \begin{align*} y\end{align*} by solving for \begin{align*} y\end{align*} first.
- Can be used to graph the line without having to rewrite the equation in slope-intercept form because a slope and a point determine a unique line. See Example 5.

Use Example 3 to show that any point on the line can be substituted for \begin{align*}(x_0,y_0)\end{align*}. Point-slope equations will simplify to the *same* slope-intercept equation regardless of the chosen point.

Use Example 6 to introduce function notation for equations in point-slope form.

- Remind students that \begin{align*}f(5.5)=12.5\end{align*} is equivalent to the ordered pair \begin{align*}(5.5, 12.5)\end{align*} in \begin{align*}6b\end{align*}.

“Flat fees”, initial amounts, starting times, etc. correspond to the intercept along the vertical axis.

## Error Troubleshooting

In Example 7, have students determine the independent and dependent variables first. This helps them form correct ordered pairs.