Find the Equation of a Line

The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. If data appears to form a straight line when graphed, an equation of that line can be used to predict the value of one variable based on the value of the other. To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line. In this section we look at several ways to write the equation of a line — the method we use is determined by what information we are given.

Find an equation of the line given the slope and $y$-intercept

If we already know the slope and $y$-intercept of a line, we can find its equation by substituting the values directly into the slope-intercept form, $y = mx + b$.

Example. Find an equation of a line with slope $-7$ and $y$-intercept $(0, -1)$.

We substitute $m = -7$ and $b = -1$ into $y = mx + b$:

Example. Find the equation of the line shown, whose $y$-intercept is $(0, -4)$ and which also passes through $(3, -2)$.

We find the slope by counting the rise and run between the two points: $m = \tfrac{2}{3}$. Substituting $m = \tfrac{2}{3}$ and $b = -4$ into $y = mx + b$:

Find an equation of the line given the slope and a point

The slope-intercept form works well when we know the slope and $y$-intercept. But what if we know the slope and some other point that isn’t the $y$-intercept? We can derive another form of the equation by starting with the slope formula for a line through a specific point $(x_1, y_1)$ and some other point $(x, y)$:

Multiplying both sides by $x - x_1$ and simplifying gives $y - y_1 = m(x - x_1)$.

We can use the point-slope form to find an equation of a line when we are given the slope and one point, then rewrite the equation in slope-intercept form.

Example. Find an equation of a line with slope $m = \tfrac{2}{5}$ that contains the point $(10, 3)$. Write the equation in slope-intercept form.

Substituting $m = \tfrac{2}{5}$ and $(x_1, y_1) = (10, 3)$ into the point-slope form:

Example. Find an equation of a horizontal line that contains the point $(-1, 2)$. Write the equation in slope-intercept form.

Every horizontal line has slope $0$. Substituting $m = 0$ and $(x_1, y_1) = (-1, 2)$:

This is the equation of the horizontal line $y = 2$.

Find an equation of the line given two points

When real-world data is collected, a linear model can often be created from just two data points. We have two options so far for finding an equation of a line: slope-intercept or point-slope. Since we will know two points rather than the slope directly, it makes more sense to first find the slope from the two points, then use the point-slope form.

Example. Find an equation of a line that contains the points $(5, 4)$ and $(3, 6)$. Write the equation in slope-intercept form.

First we find the slope:

Choosing the point $(5, 4)$ and substituting into the point-slope form:

Using the other point, $(3, 6)$, gives the same equation.

Example. Find an equation of a line that contains the points $(-2, 4)$ and $(-2, -3)$. Write the equation in slope-intercept form.

Finding the slope:

The slope is undefined, telling us this is a vertical line. Both points have $x$-coordinate $-2$, so the equation of the line is $x = -2$. Since there is no $y$, we cannot write it in slope-intercept form.

Find an equation of a line parallel to a given line

Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. Since parallel lines have the same slope, we already have the slope we need — we just combine it with the given point using the point-slope form. We write $m_\parallel$ for the slope of a line parallel to a line with slope $m$.

Example. Find an equation of a line parallel to $y = 2x - 3$ that contains the point $(-2, 1)$. Write the equation in slope-intercept form.

The given line is in slope-intercept form with $m = 2$. Since parallel lines have the same slope, $m_\parallel = 2$. Substituting $m = 2$ and $(x_1, y_1) = (-2, 1)$ into the point-slope form:

Find an equation of a line perpendicular to a given line

Now consider finding a line through a specific point that is perpendicular to a given line. Perpendicular lines have slopes that are negative reciprocals of each other, so we again already have the slope we need. We write $m_\perp$ for the slope of a line perpendicular to a line with slope $m$.

Example. Find an equation of a line perpendicular to $y = 2x - 3$ that contains the point $(-2, 1)$. Write the equation in slope-intercept form.

The given line has $m = 2$, so the perpendicular slope is the negative reciprocal, $m_\perp = -\tfrac{1}{2}$. Substituting $m = -\tfrac{1}{2}$ and $(x_1, y_1) = (-2, 1)$:

Example. Find an equation of a line perpendicular to $x = 5$ that contains the point $(3, -2)$. Write the equation in slope-intercept form.

The line $x = 5$ is vertical, so any line perpendicular to it must be horizontal, with slope $0$. Substituting $m_\perp = 0$ and $(x_1, y_1) = (3, -2)$:

Example. Find an equation of a line perpendicular to $y = -4$ that contains the point $(-4, 2)$. Write the equation in slope-intercept form.

The line $y = -4$ is horizontal, so any line perpendicular to it must be vertical, in the form $x = a$. Since the perpendicular line passes through $(-4, 2)$, every point on it has $x$-coordinate $-4$. The equation of the perpendicular line is $x = -4$.

Key terms

point-slope form — the form $y - y_1 = m(x - x_1)$ of an equation of a line with slope $m$ that contains the point $(x_1, y_1)$; useful whenever the slope and any one point (not necessarily the $y$-intercept) are known, or when two points are known and the slope is found first.

Practice

Find an equation of the line given the slope and $y$-intercept

Find an equation of the line given the slope and a point

Find an equation of the line given two points

Find an equation of a line parallel to a given line

Find an equation of a line perpendicular to a given line

This section is adapted from Elementary Algebra 2e, Section 4.6: Find the 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: condensed the worked examples and tables; omitted the Be Prepared quiz, Media links, Self Check checklist, and unselected Section Exercises; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the section-final interactive Practice block.