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 it x.  Now, we will discuss equations in which two letters are used for variables, usually we call them x and y.

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.

2x y = 4          y = 3x 6          y = 2x          5x 3y = 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 y.  Therefore, it is standard procedure to assign the x-variable to the horizontal axis, which is then called the x-axis, and the y-variable to the vertical axis, which is then called the y-axis. 

NOTE:  Sometimes equations may be expressed in terms of other variables, say, p and q.  In this case, you must be told which variable to assign to what axis of the Cartesian Coordinate System!

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

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 y = 4  , find the value for x if y = 0.  Write the information as an ordered pair.

2x 0 = 4

2x = 4

x = 2

 The ordered pair is  (2, 0).

Given the linear equation 2x y = 4  , find the value for y if x = 0.  Write the information as an ordered pair.

2(0) y = 4

y = 4

y = 4

The ordered pair is (0, 4).

Given the linear equation 2x y = 4  , find the value for y if x = 4.  Write the information as an ordered pair.

2(4) y = 4

8 y = 4

y = 4

and y = 4

The ordered pair is (4, 4).

Example 3:

Given the linear equation y = 3x 6, find the value for x if y = 0.  Write the information as an ordered pair.

0 = 3x 6

3x = 6

x = 2

The ordered pair is (2, 0).

Given the linear equation y = 3x 6, find the value for y if x = 0.  Write the information as an ordered pair.

y = 3(0) 6

y = 6

The ordered pair is (0, 6).

Given the linear equation y = 3x 6, find the value for y if 4.  Write the information as an ordered pair.

y = 3(4) 6

y = 12 6

and y = 6

The ordered pair is (4, 6).

Example 4:

Given the linear equation y = 2x , find the value for x if y = 0.  Write the information as an ordered pair.

0 = 2x

x = 0

The ordered pair is (0, 0).

Given the linear equation y = 2x, find the value for y if x = 0.  Write the information as an ordered pair.

y = 2(0)

y = 0

The ordered pair is (0, 0).

Given the linear equation y = 2x, find the value for y if x = 4 and x = 4.  Write the information as ordered pairs.

 y = 2(4)

and y = 8

The ordered pair is (4, 8).

Next, y = 2(4)

and y = 8

The second ordered pair is (4, 8).

Example 5:

Given the linear equation 5x 3y = 7, find the value for x if y = 0.  Write the information as an ordered pair.

5x 3(0) = 7

5x = 7

 The ordered pair is

Given the linear equation 5x 3y = 7, find the value for y if x = 0.  Write the information as an ordered pair.

5(0) 3y = 7

3y = 7

The ordered pair is .

Given the linear equation 5x 3y = 7 find the value for y if x = 2 and x = 1.  Write the information as ordered pairs.

5(2) 3y = 7

10 3y = 7

3y = 3

y = 1 

The ordered pair is (2, 1).

Next, 5(1) 3y = 7

5 3y = 7

3y = 12

and y =

The ordered pair is (1, 4).

Example 6:

Given the linear equation y = 2, find the value for x = 5, 1, 100.  Write the information as an ordered pair.

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

0( 5) + y = 2

y = 2

The ordered pair is (5, 2).

0( 1) + y = 2

y = 2

The ordered pair is (1, 2).

0( 100) + y = 2

y = 2

The 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 = 4, find the value for y = 1, 20, 1000.  Write the information as an ordered pair.

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

x + 0(1) = 4

x = 4

The ordered pair is (4, 1).

x + 0(20) = 4

x = 4

The ordered pair is (4, 20).

x + 0(1000) = 4

x = 4

The ordered pair is (4, 1000).

Please note that no matter what y-value is chosen, the x-value is always 4.