Lesson 4: Slope-Intercept Form

New Concept

The slope-intercept form, $y = mx + b$, is a powerful way to write linear equations. It instantly reveals a line's slope ($m$) and its $y$-intercept ($b$), making it simple to graph and interpret real-world relationships.

What’s next

Now, let's put this into action. You'll work through interactive examples to identify the slope and intercept, then tackle graphing challenges.

Property

1. The constant term tells us the vertical intercept of the graph.
2. The coefficient of the input variable tells us the slope of the graph.

Examples

• In the equation for oven temperature, $H = 0.4t + 70$, the slope is $0.4$ (temperature rise per minute) and the $y$-intercept is $70$ (the starting temperature).
• For a taxi fare modeled by $C = 2.50 + 1.50d$, the slope is $1.50$ (cost per mile) and the $y$-intercept is $2.50$ (the initial fee).
• If your phone plan is $P = 20 + 5g$, the slope is $5$ (cost per gigabyte) and the $y$-intercept is $20$ (the base monthly charge).

Explanation

Think of an equation's numbers as clues. The coefficient of the variable reveals the line's steepness (slope), while the constant term shows exactly where the line starts on the vertical axis (y-intercept).

Property

A linear equation written in the form

is said to be in slope-intercept form. The coefficient $m$ is the slope of the graph, and $b$ is the $y$-intercept.

Examples

• The equation $y = 3x + 5$ is in slope-intercept form. The slope is $3$ and the $y$-intercept is $(0, 5)$.
• For $y = -2x - 1$, the slope is $-2$ and the $y$-intercept is $(0, -1)$.
• In the equation $y = \frac{1}{2}x + 4$, the slope is $\frac{1}{2}$ and the $y$-intercept is $(0, 4)$.

Explanation

This form is a recipe for drawing a line. The '$b$' tells you your starting point on the y-axis, and the '$m$' (slope) gives you directions on how steep to draw the line from there.

Property

To Graph a Line Using the Slope-Intercept Method:

1. Write the equation in the form $y = mx + b$.
2. Plot the $y$-intercept, $(0, b)$.
3. Write the slope as a fraction, $m = \frac{\Delta y}{\Delta x}$.
4. Use the slope to find a second point on the graph: Starting at the $y$-intercept, move $\Delta y$ units in the $y$-direction, then $\Delta x$ units in the $x$-direction.
5. Find a third point by moving $-\Delta y$ units in the $y$-direction, then $-\Delta x$ units in the $x$-direction, starting from the $y$-intercept.
6. Draw a line through the three plotted points.

Examples

• To graph $y = \frac{2}{3}x + 1$, first plot the $y$-intercept at $(0, 1)$. From there, use the slope to move up 2 units and right 3 units to find a second point at $(3, 3)$.
• For $y = -3x + 5$, start by plotting $(0, 5)$. Write the slope as $\frac{-3}{1}$, then move down 3 units and right 1 unit to find the point $(1, 2)$.
• To graph $y = \frac{5}{2}x - 3$, plot the intercept at $(0, -3)$. From that point, move up 5 units and right 2 units to locate a second point at $(2, 2)$.

Explanation

This method is like a treasure map for graphing. Start at 'X marks the spot' on the y-axis (the intercept, $b$), then follow the slope's 'rise over run' directions to find your next point and draw the line.

Property

We can write the equation of any non-vertical line in slope-intercept form by solving the equation for $y$ in terms of $x$.

Caution: Do not confuse solving for $y$ with finding the $y$-intercept. When we 'solve for $y$', we are writing the equation in another form, so both variables, $x$ and $y$, still appear in the equation.

Examples

• To convert $6x + 3y = 12$, subtract $6x$ from both sides to get $3y = -6x + 12$. Then divide all terms by $3$ to get the final form $y = -2x + 4$.
• For the equation $5x - 2y = 10$, subtract $5x$ to get $-2y = -5x + 10$. Divide everything by $-2$ to find the slope-intercept form, $y = \frac{5}{2}x - 5$.
• To solve $x + 4y = 8$ for $y$, subtract $x$ from both sides giving $4y = -x + 8$. Then divide by $4$ to get $y = -\frac{1}{4}x + 2$.