Learning Outcomes

Determine if a graph or equation is linear
Create a linear equation for a given scenario
For a given linear equation, identify the independent and dependent variables
Interpret the slope and y-intercept in context for a given linear equation

Recall: GRAPHING INDEPENDENT & DEPENDENT VARIABLE

When both the input (independent variable) and the output (dependent variable) are real numbers, a function can be represented by a coordinate graph. The input is plotted on the horizontal x-axis and the output is plotted on the vertical y-axis.

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

[latex]y = a + bx[/latex]

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 1

The following examples are linear equations.

[latex]y = 3 + 2x[/latex]

[latex]y = -0.01 + 1.2x[/latex]

try it 1

Is the following an example of a linear equation?

[latex]y = –0.125 – 3.5x[/latex]

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

Example 2

Graph the equation [latex]y = –1 + 2x[/latex].

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.

Recall: GRAPHING A LINEAR EQUATION OF THE FORM [latex]y = a + bx[/latex]

To graph an equation in the form of [latex]y = a + bx[/latex], the a is where the line crosses the [latex]y[/latex]-axis and the [latex]b[/latex] is the slope. Slope is the [latex]\frac{\mathrm{rise}}{\mathrm{run}}[/latex]. The rise goes up for a positive number and down for negative numbers, the run goes right with a positive number and left with negative numbers. So to graph the line in Example 2, put a dot where the equation crosses the [latex]y[/latex]-axis (at -1) and count up 2 and right 1.

try it 2

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

This is a graph of an equation. The x-axis is labeled in intervals of 2 from 0 - 14; the y-axis is labeled in intervals of 2 from 0 - 12. The equation's graph is a curve that crosses the y-axis at 2 and curves upward and to the right.

Example 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.

Show Answer

try it 3

Emma’s Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class as well as $20 per student in the class. The total cost Emma pays depends on the number of students in a class. Find the equation that expresses the total cost in terms of the number of students in a class.

Slope and Y-Intercept of a Linear Equation

For the linear equation y = a + bx, b = slope and a = y-intercept. From algebra, recall that the slope is a number that describes the steepness of a line, and the y-intercept is the y coordinate of the point (0, a) where the line crosses the y-axis.

Three possible graphs of the equation y = a + bx. For the first graph, (a), b > 0 and so the line slopes upward to the right. For the second, b = 0 and the graph of the equation is a horizontal line. In the third graph, (c), b < 0 and the line slopes downward to the right.

Three possible graphs of y = a + bx. (a) If b > 0, the line slopes upward to the right. (b) If b = 0, the line is horizontal. (c) If b < 0, the line slopes downward to the right.

Example 4

Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is y = 25 + 15x.

What are the independent and dependent variables? What is the y-intercept and what is the slope? Interpret them using complete sentences.