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5.1: Linear Equations in Slope-Intercept Form

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

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

• Write an equation given slope and \begin{align*}y–\end{align*}intercept.
• Write an equation given the slope and a point.
• Write an equation given two points.
• Write a linear function in slope-intercept form.
• Solve real-world problems using linear models in slope-intercept form.

Vocabulary

Terms introduced in this lesson:

slope-intercept form

Teaching Strategies and Tips

Slope-intercept form:

• Allows quick identification of the slope and \begin{align*}y-\end{align*}intercept. This makes graphing linear functions easier, for example.
• Written so that the dependent variable is isolated. This makes finding ordered pairs easier, for example.

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

• The slope and \begin{align*}y-\end{align*}intercept of the line. See Examples 1, 2, and 7.
• The slope and any one point on the line (\begin{align*}b\end{align*} is not given). See Examples 3 and 8.
• Any two points on the line (neither \begin{align*}m\end{align*} nor \begin{align*}b\end{align*} are given). See Examples 4 and 9.

In Example 2, have students check their answer by choosing two different points on each line and finding the slope. Construct triangles using lattice points.

In Example 4, suggest that students label their ordered pairs, writing \begin{align*}x_1,y_1\end{align*} and \begin{align*}x_2, y_2\end{align*} above each coordinate.

Use Example 5 to demonstrate plugging an expression into a function. Encourage students to keep the parentheses with the expression as it gets plugged into the function.

Let \begin{align*}f(x)=-2x+3\end{align*}. Find \begin{align*}f(-4x+1)\end{align*}.

Solution: The entire expression \begin{align*}(-4x+1)\end{align*} must be plugged into the function. Keep the parentheses as you plug in:

\begin{align*}f((-4x+1)) & = -2(-4x+1)+3 \\ f((-4x+1)) & = 8x-2+3=8x+1\end{align*}

Check for student conceptual understanding of function notation. For example,

• \begin{align*}f(-2)=5\end{align*} is equivalent to the ordered pair \begin{align*}(-2,5)\end{align*} or \begin{align*}x = -2\end{align*} and \begin{align*}y = 5\end{align*}.
• \begin{align*}f(0)=5\end{align*} is a \begin{align*}y-\end{align*}intercept. \begin{align*}f(3)=0\end{align*} is an \begin{align*}x-\end{align*}intercept.

Assume that Examples 7-9 can be modeled by a linear function.

• The slope and \begin{align*}y-\end{align*}intercept have contextual significance in applied problems. Have students interpret each in context of the problem.
• In applied problems, students can single out the slope from among the givens by looking for key phrases that signify a slope. For example, dollars per hour (Example 7) and inches per day (Example 8).

Error Troubleshooting

In Example 2, remind students that a run left or a rise down are negative quantities. Students commit the error in part a. for example when they go down \begin{align*}1\end{align*}, right \begin{align*}1\end{align*} but write the slope as \begin{align*}+1\end{align*} instead of \begin{align*}-1\end{align*}.

In Example 3b, remind students to find common denominators.

In Example 4b, remind students to watch the double negative in the denominator: \begin{align*}\frac{3-1}{-2-(-4)}\end{align*}.

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