Lesson 5: Equations of Lines

New Concept

Explore how real-world data, like the effect of salinity on water's freezing point, can be modeled using linear equations. We'll learn to write, graph, and interpret these equations to understand and predict important trends.

What’s next

Next, you’ll work through interactive examples and practice cards to master writing the equation of a line.

Property

The slope-intercept form for a linear equation is $y = mx + b$, where $m$ is the slope of the line and the point $(0, b)$ is the y-intercept.

The slope tells us the rate of change, and the y-intercept is where the line crosses the vertical axis.

Examples

Find the equation of a line with slope 3 and y-intercept at $(0, -5)$. Using the slope-intercept form $y = mx + b$, we substitute $m=3$ and $b=-5$ to get the equation $y = 3x - 5$.

Property

The point-slope form of a linear equation is $y = y_1 + m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is any known point on the line.

This form is useful for finding the equation of a line when you know its slope and at least one point it passes through.

You may also see the formula written in an alternate version:

Property

To find the equation of a line passing through two points, $(x_1, y_1)$ and $(x_2, y_2)$, first calculate the slope using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Then, use this slope and one of the points with the point-slope form, $y - y_1 = m(x - x_1)$, to write the equation.

Examples

Find the equation of the line passing through $(2, 5)$ and $(4, 9)$. First, find the slope: $m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2$. Using point-slope form with $(2, 5)$, we get $y - 5 = 2(x - 2)$.

What is the equation of the line containing points $(-1, 6)$ and $(3, -2)$? The slope is $m = \frac{-2 - 6}{3 - (-1)} = \frac{-8}{4} = -2$. Using point $(-1, 6)$, the equation is $y - 6 = -2(x + 1)$.