** LINEAR EQUATIONS IN TWO VARIABLES AND THE
CARTESIAN COORDINATE SYSTEM**

**Copyright by Ingrid Stewart, Ph.D. Please Send
Questions and Comments to ingrid.stewart@csn.edu.**

Equations in Two Variables

Up to this point, we discusses equations in one variable. For example, all of the linear equations we solved contained exactly one variable. Most of the time we called itx. Now, we will discuss equations in which two letters are used for variables, usually we call themxandy.

Following are a few examples of LINEAR equations in two variable. We know that they are linear equations because both variables are raised to the power of 1.

2xy = 4y =3x6y = 2x5x3y = 7

y = 2 x =4

NOTE: The last four equations look different from the previous four examples. There is only one variable in each one!

Often we want to show a pictorial representation of equations in two variables to give us a better understanding of some of the characteristics of an equation. We do this with the help of the

Cartesian Coordinate System.

The Cartesian Coordinate System

The
*Cartesian Coordinate System*
must have two intersecting axes. A horizontal one and a vertical
one. In mathematics, most equations are expressed in terms of** x** and

NOTE:
Sometimes equations may be expressed in terms of other variables, say, ** p** and

The axes divide the coordinate plane into four areas. We use Roman numerals and call them Quadrant I, Quadrant II, Quadrant III, and Quadrant IV.

The
point of intersection of the two axes is called the **Origin**.

Each
axis must be partitioned into identical units using hash marks. All of
the hash marks MUST be numbered to indicate the "scale" of the axis.
The "scale" always depends on the type of equation that is to be
graphed.

NOTE:
The units partitioning the x-axis do not have to be equal in length to the
units partitioning the y-axis. As a matter of fact, given limited
graphing space, it is often necessary and desirable to have differing scales
along the x- and y-axes.

The
location of a point in the *Cartesian Coordinate System* is indicated by
finding the number of units the point is removed from the *Origin* both in
the x- and y-axis direction. These two numbers are then placed into
parentheses separated by a comma. We call one number the *x-coordinate*
and the other number the *y-coordinate*. The x-coordinate is always
stated before the y-coordinate. Both coordinates together are called an *ordered pair*.

Example 1:Plot the following points into a

Cartesian Coordinate System.

(4,,2)(,3, 2)(0,,3), and(2, 0). The coordinates(0, 0)are reserved for the Origin.(0,0)

Please note that it is common to use the number representing the x-coordinate first and move horizontally along the x-axis either in the positive or negative direction depending on the value of the x-coordinate.

From the location of the x-coordinate, we will then move in a vertical direction up (positive) or down (negative) using the number representing the y-coordinate.

Example 2:Given the linear equation

2x, find the value fory = 4ifx. Write the information as an ordered pair.y = 0

2x0 = 4

2x = 4

x = 2The ordered pair is

.(2, 0)Given the linear equation

2x, find the value fory = 4ify. Write the information as an ordered pair.x = 0

2(0)y = 4

y = 4

y =4The ordered pair is

(0,.4)Given the linear equation

2x, find the value fory = 4ify. Write the information as an ordered pair.x = 4

2(4)y = 4

8y = 4

y = 4and

y = 4The ordered pair is

.(4, 4)

Example 3:Given the linear equation

y =3x, find the value for6ifx. Write the information as an ordered pair.y = 0

0 =3x6

3x =6

x =2The ordered pair is

(.2, 0)Given the linear equation

y =3x, find the value for6ify. Write the information as an ordered pair.x = 0

y =3(0)6

y =6The ordered pair is

(0,.6)Given the linear equation

y =3x, find the value for6ify. Write the information as an ordered pair.4

y =3(4)6

y = 126and

y = 6The ordered pair is

(.4, 6)

Example 4:Given the linear equation

, find the value fory = 2xifx. Write the information as an ordered pair.y = 0

0 = 2x

x = 0The ordered pair is

.(0, 0)Given the linear equation

, find the value fory = 2xify. Write the information as an ordered pair.x = 0

y = 2(0)

y = 0The ordered pair is

.(0, 0)Given the linear equation

, find the value fory = 2xifyandx = 4x =. Write the information as ordered pairs.4

y = 2(4)

andy = 8The ordered pair is

.(4, 8)Next,

y = 2(4)and

y =8The second ordered pair is

(4,.8)

Example 5:Given the linear equation

5x, find the value for3y = 7ifx. Write the information as an ordered pair.y = 0

5x3(0) = 7

5x = 7

The ordered pair is

Given the linear equation

5x, find the value for3y = 7ify. Write the information as an ordered pair.x = 0

5(0)3y = 7

3y = 7

The ordered pair is .

Given the linear equation

5xfind the value for3y = 7ifyandx = 2x =. Write the information as ordered pairs.1

5(2)3y = 7

103y = 7

3y =3

y = 1The ordered pair is

.(2, 1)Next,

5(1)3y = 7

53y = 7

3y = 12and

y =4The ordered pair is

(1,.4)

Example 6:Given the linear equation

, find the value fory = 2x =. Write the information as an ordered pair.5, 1, 100

NOTE: We can think ofy = 2as an equation in two variables by rewriting the equation as.0x + y = 2

0(5) + y = 2

y = 2The ordered pair is

(.5, 2)

0( 1) + y = 2

y = 2The ordered pair is

.(1, 2)

0( 100) + y = 2

y = 2The ordered pair is

.(100, 2)

Please note that no matter what x-value is chosen, the y-value is always 2.

Example 7:Given the linear equation

x =, find the value for4y =. Write the information as an ordered pair.1, 20, 1000

NOTE: We can think ofx =4as an equation in two variables by rewriting the equation asx + 0y =.4

x + 0(1) =4

x =4The ordered pair is

(4,1.)

x + 0(20) =4

x =4The ordered pair is

(.4, 20)

x + 0(1000) =4

x =4The ordered pair is

(.4, 1000)

Please note that no matter what y-value is chosen, the x-value is always4.