Lesson 4.5: Use the Slope-Intercept Form of an Equation of a Line

New Concept

The slope-intercept form, $y = mx + b$, provides a powerful shortcut to understanding and graphing linear equations. It instantly reveals a line's slope ($m$) and y-intercept ($b$), making it easier to visualize lines and analyze their relationships.

What’s next

Next, you'll work through interactive examples to master graphing with this form and then solve challenge problems involving parallel and perpendicular lines.

Property

The slope-intercept form of an equation of a line with slope $m$ and y-intercept, $(0, b)$, is

Examples

• Identify the slope and y-intercept for the line $y = -2x + 8$. The equation is in $y=mx+b$ form. The slope $m$ is $-2$, and the y-intercept is at $(0, 8)$.

• For the equation $3x + y = 7$, first solve for $y$ to get $y = -3x + 7$. Now you can see the slope $m$ is $-3$ and the y-intercept is at $(0, 7)$.

Property

To graph a line using its slope and intercept:

1. Find the slope-intercept form of the equation, $y = mx + b$.
2. Identify the slope ($m$) and y-intercept ($b$).
3. Plot the y-intercept at point $(0, b)$.
4. Use the slope formula $m = \frac{\text{rise}}{\text{run}}$ to find a second point by counting from the y-intercept.
5. Connect the two points with a straight line.

Examples

• To graph $y = 3x - 1$, start by plotting the y-intercept at $(0, -1)$. The slope $m=3$ means $\frac{\text{rise}}{\text{run}} = \frac{3}{1}$. From $(0, -1)$, go up 3 units and right 1 unit to find the next point, $(1, 2)$. Connect them.

• To graph $y = -\frac{1}{2}x + 3$, plot the y-intercept at $(0, 3)$. The slope $m=-\frac{1}{2}$ means $\frac{\text{rise}}{\text{run}} = \frac{-1}{2}$. From $(0, 3)$, go down 1 unit and right 2 units to find the point $(2, 2)$. Connect them.

Property

To determine the most convenient method to graph a line:

• For an equation like $y = c$ (e.g., $y=5$), recognize it as a horizontal line passing through the y-axis at $c$.
• For an equation like $x = c$ (e.g., $x=-2$), recognize it as a vertical line passing through the x-axis at $c$.
• For an equation in $Ax + By = C$ form (e.g., $2x+3y=6$), finding the x- and y-intercepts is often easiest.
• For an equation in $y = mx + b$ form (e.g., $y = 2x-1$), using the slope and y-intercept is most direct.

Examples

• For the equation $y = 5$, the most convenient method is to draw a horizontal line that passes through the y-axis at 5.

• For the equation $4x - 3y = 12$, finding the intercepts is convenient. If $x=0$, $y=-4$. If $y=0$, $x=3$. Plot $(0, -4)$ and $(3, 0)$ and connect them.

Property

Parallel lines are lines in the same plane that do not intersect.

• Parallel lines have the same slope and different y-intercepts.
• If $m_1$ and $m_2$ are the slopes of two parallel lines, then $m_1 = m_2$.
• Parallel vertical lines have different x-intercepts.

Examples

• The lines $y = 5x + 1$ and $y = 5x - 3$ are parallel because they both have a slope of $m=5$ but have different y-intercepts.

• To check if $2x + y = 7$ and $y = -2x + 4$ are parallel, rewrite the first equation as $y = -2x + 7$. Both lines have a slope of $m=-2$ and different y-intercepts, so they are parallel.

Property

Perpendicular lines are lines in the same plane that form a right angle.

• If $m_1$ and $m_2$ are the slopes of two perpendicular lines, then their slopes are negative reciprocals of each other: $m_1 \cdot m_2 = -1$ and $m_1 = -\frac{1}{m_2}$.
• Vertical lines and horizontal lines are always perpendicular to each other.

Examples

• The lines $y = 4x + 2$ and $y = -\frac{1}{4}x + 5$ are perpendicular. Their slopes, $4$ and $-\frac{1}{4}$, are negative reciprocals, since $4 \cdot (-\frac{1}{4}) = -1$.

• To check if $x - 3y = 9$ and $y = -3x + 1$ are perpendicular, rewrite the first equation as $y = \frac{1}{3}x - 3$. The slopes are $\frac{1}{3}$ and $-3$. Since $\frac{1}{3} \cdot (-3) = -1$, the lines are perpendicular.