Use the Slope-Intercept Form of an Equation of a Line

Recognize the relation between the graph and the slope-intercept form of an equation of a line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point-slope method. Once we see how an equation in slope-intercept form and its graph are related, we’ll have one more method we can use to graph lines.

Earlier we graphed the line of the equation $y = \tfrac{1}{2}x + 3$ by plotting points. Let’s find the slope of this line the way we did in the previous section — using two points from the graph.

The rise is $1$ and the run is $2$. Substituting into the slope formula:

What is the $y$-intercept of the line? The $y$-intercept is where the line crosses the $y$-axis, so the $y$-intercept is $(0, 3)$. The equation of this line is $y = \tfrac{1}{2}x + 3$. Notice that the line has slope $m = \tfrac{1}{2}$ and $y$-intercept $(0, 3)$.

When a linear equation is solved for $y$, the coefficient of the $x$ term is the slope and the constant term is the $y$-coordinate of the $y$-intercept. We say that the equation $y = \tfrac{1}{2}x + 3$ is in slope-intercept form.

Example. Use the graph to find the slope and $y$-intercept of the line $y = 2x + 1$, and compare these values to the equation $y = mx + b$.

To find the slope of the line, we choose two points on the line, $(0, 1)$ and $(1, 3)$. The rise is $2$ and the run is $1$, so:

The $y$-intercept is the point $(0, 1)$. We found slope $m = 2$ and $y$-intercept $(0, 1)$, matching the equation $y = 2x + 1$: the slope is the same as the coefficient of $x$, and the $y$-coordinate of the $y$-intercept is the same as the constant term.

Identify the slope and $y$-intercept from an equation of a line

When we are given an equation in slope-intercept form, we can use the $y$-intercept as a point, and then count out the slope from there. Let’s practice finding the values of the slope and $y$-intercept from the equation of a line.

Example. Identify the slope and $y$-intercept of the line with equation $y = -3x + 5$.

We compare the equation to the slope-intercept form $y = mx + b$: the slope is $m = -3$, and the $y$-intercept is $(0, 5)$.

When an equation of a line is not given in slope-intercept form, our first step will be to solve the equation for $y$.

Example. Identify the slope and $y$-intercept of the line with equation $x + 2y = 6$.

This equation is not in slope-intercept form. To compare it to the slope-intercept form, we first solve the equation for $y$:

Now the equation is in slope-intercept form $y = mx + b$, so we can identify the slope, $m = -\tfrac{1}{2}$, and the $y$-intercept, $(0, 3)$.

Graph a line using its slope and intercept

Now that we know how to find the slope and $y$-intercept of a line from its equation, we can graph the line by plotting the $y$-intercept and then using the slope to find another point.

Example. Graph the line of the equation $y = 4x - 2$ using its slope and $y$-intercept.

The equation is already in slope-intercept form: $y = mx + b$, so $m = 4$ and the $y$-intercept is $(0, -2)$. We plot $(0, -2)$. The slope is $m = 4 = \tfrac{4}{1}$, so the rise is $4$ and the run is $1$. Starting at $(0, -2)$, we count up $4$ and right $1$ to mark the second point, $(1, 2)$, then connect the two points with a line.

To check our work, we can find another point on the line and make sure it is a solution of the equation. The graph also passes through $(4, 0)$: substituting into $y = -x + 4$ — wait, let’s check $(1,2)$ in $y = 4x - 2$ instead: $2 \stackrel{?}{=} 4(1) - 2$, so $2 = 2$. ✓

Choose the most convenient method to graph a line

Now that we have seen several methods to graph lines, how do we know which method to use for a given equation? While we could plot points, use the slope-intercept form, or find the intercepts for any equation, recognizing the most convenient way to graph a certain type of equation makes our work easier. Generally, plotting points is not the most efficient way to graph a line.

Here are six equations and the method used to graph each of them:

Equations with just one variable have graphs that are vertical or horizontal lines. If both $x$ and $y$ are on the same side of the equation — of the form $Ax + By = C$ — we substitute $y = 0$ to find the $x$-intercept and $x = 0$ to find the $y$-intercept, and then find a third point. Equations already written in slope-intercept form are graphed fastest by identifying the slope and $y$-intercept directly.

Example. Determine the most convenient method to graph each line: (a) $y = -6$ (b) $5x - 3y = 15$ (c) $x = 7$ (d) $y = \tfrac{2}{5}x - 1$.

(a) This equation has only one variable, $y$. Its graph is a horizontal line crossing the $y$-axis at $-6$.

(b) This equation is of the form $Ax + By = C$. The easiest way to graph it will be to find the intercepts and one more point.

(c) There is only one variable, $x$. The graph is a vertical line crossing the $x$-axis at $7$.

(d) Since this equation is in $y = mx + b$ form, it will be easiest to graph this line by using the slope and $y$-intercept.

Graph and interpret applications of slope-intercept

Many real-world applications are modeled by linear equations. Usually when a linear equation models a real-world situation, different letters are used for the variables instead of $x$ and $y$ — the variable names remind us of what quantities are being measured.

Example. The equation $F = \tfrac{9}{5}C + 32$ is used to convert temperatures, $C$, on the Celsius scale to temperatures, $F$, on the Fahrenheit scale.

(a) Find the Fahrenheit temperature for a Celsius temperature of $0$.

(b) Find the Fahrenheit temperature for a Celsius temperature of $20$.

(c) Interpret the slope and $F$-intercept of the equation.

(d) Graph the equation.

(a) Find $F$ when $C = 0$: $F = \tfrac{9}{5}(0) + 32 = 32$.

(b) Find $F$ when $C = 20$: $F = \tfrac{9}{5}(20) + 32 = 36 + 32 = 68$.

(c) Even though this equation uses $F$ and $C$, it is still in slope-intercept form. Comparing $F = mC + b$ to $F = \tfrac{9}{5}C + 32$: the slope, $\tfrac{9}{5}$, means that the Fahrenheit temperature increases $9$ degrees when the Celsius temperature increases $5$ degrees. The $F$-intercept means that when the temperature is $0°$ on the Celsius scale, it is $32°$ on the Fahrenheit scale.

(d) To graph the equation we start at the $F$-intercept $(0, 32)$, then count out the rise of $9$ and the run of $5$ to get a second point.

Example. Stella has a home business selling gourmet pizzas. The equation $C = 4p + 25$ models the relation between her weekly cost, $C$, in dollars, and the number of pizzas, $p$, that she sells.

(a) Find Stella’s cost for a week when she sells no pizzas: $C = 4(0) + 25 = 25$. Her fixed cost is $\text{\textdollar}25$ when she sells no pizzas.

(b) Find the cost for a week when she sells $15$ pizzas: $C = 4(15) + 25 = 85$. Her costs are $\text{\textdollar}85$ when she sells $15$ pizzas.

(c) Interpret the slope and $C$-intercept: comparing $C = mp + b$ to $C = 4p + 25$, the slope, $4$, means that the cost increases by $\text{\textdollar}4$ for each pizza Stella sells. The $C$-intercept means that even when Stella sells no pizzas, her costs for the week are $\text{\textdollar}25$.

(d) To graph the equation, start at the $C$-intercept $(0, 25)$, then count out the rise of $4$ and the run of $1$ to get a second point.

Use slopes to identify parallel lines

The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never intersect.

We say this more formally in terms of the rectangular coordinate system: two lines that have the same slope and different $y$-intercepts are called parallel lines.

What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit the definition above. We say that vertical lines with different $x$-intercepts are parallel.

Since parallel lines have the same slope and different $y$-intercepts, we can look at the slope-intercept form of the equations of two lines and decide whether the lines are parallel — without graphing them.

Example. Use slopes and $y$-intercepts to determine if the lines $3x - 2y = 6$ and $y = \tfrac{3}{2}x + 1$ are parallel.

We solve the first equation for $y$:

The second equation, $y = \tfrac{3}{2}x + 1$, is already in slope-intercept form. Both lines have slope $m = \tfrac{3}{2}$. The first line has $y$-intercept $(0, -3)$ and the second has $y$-intercept $(0, 1)$. The lines have the same slope and different $y$-intercepts, so they are parallel.

Example. Use slopes and $y$-intercepts to determine if the lines $y = -4$ and $y = 3$ are parallel.

Since there is no $x$-term, we write each as $y = 0x - 4$ and $y = 0x + 3$. Both lines have slope $m = 0$; the $y$-intercepts are $(0, -4)$ and $(0, 3)$. The lines have the same slope and different $y$-intercepts, so they are parallel. (You may recognize these right away as horizontal lines, which are always parallel to each other unless they are the same line.)

Example. Use slopes and $y$-intercepts to determine if the lines $x = -2$ and $x = -5$ are parallel.

Since there is no $y$, these equations cannot be put in slope-intercept form. But we recognize them as equations of vertical lines, with $x$-intercepts $-2$ and $-5$. Since their $x$-intercepts are different, the vertical lines are parallel.

Example. Use slopes and $y$-intercepts to determine if the lines $y = 2x - 3$ and $-6x + 3y = -9$ are parallel.

The first equation is already in slope-intercept form: $y = 2x - 3$. We solve the second equation for $y$:

The lines have the same slope, but they also have the same $y$-intercept, $(0, -3)$. Their equations represent the same line — they are not parallel; they are the same line.

Use slopes to identify perpendicular lines

Let’s look at the lines whose equations are $y = \tfrac{1}{4}x - 1$ and $y = -4x + 2$.

These lines lie in the same plane and intersect in right angles. We call these lines perpendicular.

As we read from left to right, the line $y = \tfrac{1}{4}x - 1$ rises, so its slope is positive. The line $y = -4x + 2$ drops from left to right, so it has a negative slope. Does it make sense that the slopes of two perpendicular lines have opposite signs?

The slope of the first line, $m_1 = \tfrac{1}{4}$, and the slope of the second line, $m_2 = -4$, are negative reciprocals of each other. If we multiply them, their product is $-1$:

This is always true for perpendicular lines.

We find the slope-intercept form of each equation, and then check whether the product of the slopes is $-1$. Perpendicular lines may have the same $y$-intercepts.

Example. Use slopes to determine if the lines $y = -5x - 4$ and $x - 5y = 5$ are perpendicular.

The first equation is already in slope-intercept form: $m_1 = -5$. We solve the second equation for $y$:

so $m_2 = \tfrac{1}{5}$. The slopes are negative reciprocals of each other, so the lines are perpendicular. We check: $m_1 \cdot m_2 = -5 \left(\tfrac{1}{5}\right) = -1$. ✓

Example. Use slopes to determine if the lines $7x + 2y = 3$ and $2x + 7y = 5$ are perpendicular.

Solving both equations for $y$: $y = -\tfrac{7}{2}x + \tfrac{3}{2}$ gives $m_1 = -\tfrac{7}{2}$, and $y = -\tfrac{2}{7}x + \tfrac{5}{7}$ gives $m_2 = -\tfrac{2}{7}$. The slopes are reciprocals of each other, but they have the same sign. Since they are not negative reciprocals, the lines are not perpendicular.

Key terms

slope-intercept form — the form $y = mx + b$ of an equation of a line, where $m$ is the slope and $(0, b)$ is the $y$-intercept. parallel lines — lines in the same plane that do not intersect; they have the same slope and different $y$-intercepts (or, for vertical lines, different $x$-intercepts). perpendicular lines — lines in the same plane that form a right angle; the product of their slopes is $-1$, so their slopes are negative reciprocals of each other.

This section is adapted from Elementary Algebra 2e, Section 4.5: Use the Slope-Intercept Form of an Equation of a Line by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the slope-intercept, parallel-lines, and perpendicular-lines graphs as accessible inline graphics; condensed the worked examples and tables; omitted the Be Prepared quiz, Media links, Self Check checklist, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.