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Linear Equations

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module provides an overview of Linear Regression and Correlation: Linear Equations as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

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

y = a + bx y=a+bx
(1)

where aa and bb are constant numbers.

xx is the independent variable, and yy is the dependent variable. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.

Example 1

The following examples are linear equations.

y=3+2xy=3+2x
(2)
y=-0.01+1.2xy=-0.01+1.2x
(3)

The graph of a linear equation of the form y = a + bx y=a+bx is a straight line. Any line that is not vertical can be described by this equation.

Example 2

Figure 1: Graph of the equation y = -1 + 2x y=-1+2x.
Graph of the equation y = -1 + 2x. This is a straight line that crosses the y-axis at -1 and is sloped up and to the right, rising 2 units for every one unit of run.

Linear equations of this form occur in applications of life sciences, social sciences, psychology, business, economics, physical sciences, mathematics, and other areas.

Example 3

Aaron's Word Processing Service (AWPS) does word processing. Its rate 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 do the word processing job.

Problem 1

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

Solution

Let xx = the number of hours it takes to get the job done.

Let yy = the total cost to the customer.

The $31.50 is a fixed cost. If it takes xx hours to complete the job, then (32)(x)(32)(x) is the cost of the word processing only. The total cost is:

y=31.50+32xy=31.50+32x

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