# 12.2: Linear Equations

Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:

$y = a + b\text{x}$

where $$a$$ and $$b$$ are constant numbers.

The variable $$x$$ is the independent variable, and $$y$$ is the dependent variable. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.

Example $$\PageIndex{1}$$

The following examples are linear equations.

$y = 3 + 2\text{x}$

$y = -0.01 + 1.2\text{x}$

Exercise $$\PageIndex{1}$$

Is the following an example of a linear equation?

$y = -0.125 - 3.5\text{x}$

yes

The graph of a linear equation of the form $$y = a + b\text{x}$$ is a straight line. Any line that is not vertical can be described by this equation.

Example $$\PageIndex{2}$$

Graph the equation $$y = -1 + 2\text{x}$$.

Figure 12.2.1.

Exercise $$\PageIndex{2}$$

Is the following an example of a linear equation? Why or why not?

Figure 12.2.2.

No, the graph is not a straight line; therefore, it is not a linear equation.

Example $$\PageIndex{3}$$

Aaron's Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour plus a$31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.

Find the equation that expresses the total cost in terms of the number of hours required to complete the job.

Let $$x =$$ the number of hours it takes to get the job done.
Let $$y =$$ the total cost to the customer.