# Linear Equations and Linear Functions

It will be shown in the next section that when x is the
temperature in

Celsius degrees and y is the temperature in Fahrenheit degrees,

the numbers x and y satisfy the equation 9x-5y =-160.

From the graph below, it appears that a straight-line relationship

exists between these two quantities.

In fact, 9x-5y =-160 is called a linear equation in two variables.

**Definition:** Any equation that can be put in the form Ax+By = C,

where A, B, and C are real numbers and A and B are not both 0,

is called a linear equation in two variables. The graph of any equation

of this form is a straight line (that is why these equations are called
”linear”).

The form Ax+By = C is called the standard form of a linear equation in

two variables.

An ordered pair (x, y) whose coordinates satisfy an equation in two variables

is called a solution of the equation. The graph of an equation in two variables

is formed by graphing the set of all solutions to the equation.

**Example 1 **Find the intercepts of 3x-4y = 18. Sketch the graph.

**Example 2** Find the intercepts of the equation 2x-3y = 0. Sketch the
graph.

**Graphing Horizontal and Vertical Lines**

**Example 3** Find the intercepts of the line y = 3. Sketch the graph.

Equations of horizontal lines have the form y = k,

where k is the y-coordinate of every point on the line.

**Example 4** Find the intercepts of the line x = 2. Sketch the graph.

Equations of vertical lines have the form x = k,

where k is the x-coordinate of every point on the line.

All linear equations except for equations of the form x = k are functions.

All linear equations of the form Ax+By = C are functions if B≠0.

**Example 5** Assume that B≠0 and solve the linear
equation Ax+By = C for y.

**A linear function** is a function that can be written in the form

where m and b a real numbers. The graph of a linear function is a nonvertical line.

**Example 6** Compare each of the following functions to
and indicate

which ones are linear.

**Example 7** Tanya works for Prentice Hall as a textbook representative.

The linear function

describes her annual income I when she has total sales s.

(a) What is the implied domain of this linear

function?

(b) What is I(0)? What does this mean?

(c) What is Tanya’s salary if she sells $500,000

in books for the year?

d) Graph the function

(e) At what level of sales will Tanya’s income be

$45,000?