Equations of a Line

An equation in two first-degree variables, such as has a line as its graph, so it is called a linear equation. In the rest of this section, we consider various forms of the equation of a line. 4 x + 7 y = 20, has a line as its graph, so it is called a linear equation. In the rest of this section, we consider various forms of the equation of a line.

Example

Equation of a Line

Find the equation of the line through (0, -3) with slope 3/4.

Solution

We can use the definition of slope, letting (x₁, y₁) = (0, -3) and (x, y) represent another point on the line.

A generalization of the method of Example 2 can be used to find the equation of any line, given its y-intercept and slope. Assume that a line has y-intercept b, so that it goes through the point (0, b). Let the slope of the line be represented by m. If (x, y) is any point on the line other than (0, b) then the definition of slope can be used with the points (0, b) and (x, y) to get

This result is called the slope-intercept form of the equation of a line, because b is the y-intercept of the graph of the line.

Slope-intercept form

If a line has slope m and y -intercept b , then the equation of the line in slope-intercept form is

y = mx + b

When b = 0 we say that y is proportional to x .

Example

Slope-Intercept Form

Find the equation of the line in slope-intercept form having y -intercept 7/2 and slope -5/2

Solution

Use the slope-intercept form with b = 7/2 and m = -7/2.

The slope-intercept form shows that we can find the slope of a line by solving its equation for y. In that form the coefficient of x is the slope and the constant term is the y-intercept. For instance, in Example 2 the slope of the line 3x = 4y + 12 was given as 3/4. This slope also could be found by solving the equation for y.

4y + 12 = 3x

4y = 3x - 12

The coefficient of x, 3/4, is the slope of the line. The y-intercept is -3.