1.   Introduction to Equations in Two Variables 

Up to this point, we discussed equations in one variable, specifically they were linear equations.  Most of the time we called the variable x.  For example, .

Now, we will discuss equations in which we have two variables, usually we call them x and y.  There are infinitely many different equations in two variable.  However, the only ones we are going to discuss in this course are linear equations in two variables. You recognize them because their x and y-values are to the power of 1. For example, .

There are two other linear equations, namely horizontal and vertical lines. They are often called constant equations. For example, is a vertical line. Notice that there is NO y-value. This means that y can take on any and all real numbers. On the other hand, is a horizontal line. Notice that there is NO x-value. This means that x can take on any and all real numbers.

2.   The Cartesian Coordinate System

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 call them graphs. We make them with the help of the Cartesian Coordinate System.  It is also sometimes called a Rectangular 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!

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.  In the picture below, the hash marks on both axes are a distance of 2 units from each other.

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 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 point in the Cartesian Coordinate System corresponds to an ordered pair of real numbers .  The two numbers MUST be enclosed within parentheses!  The first number in the pair is called the x-coordinate and the second number is called the y-coordinate.

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

The x-coordinate denotes the distance and direction from the origin along the x-axis.  The y-coordinate denotes the distance and direction from the origin along the y-axis.

Example 1:

Plot the following points into a Cartesian Coordinate System.

,, , , and

We use the numbers representing the x-coordinate first and, starting at the origin, 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.

Following are the 5 points in a coordinate cystem.

Example 2:

Given the linear equation , find the value for y when .  Write the information as an ordered pair.

3(0) + y = 6

y = 6

The ordered pair is (0, 6).

Example 3:

Given the linear equation , find the value for y when .  Write the information as an ordered pair.

3(4) + y = 6

12 + y = 6

y = 6

The ordered pair is (4, 6).

Given the linear equation , find the value for x  when .  Write the information as an ordered pair

3x + 0 = 6

3x = 6

x = 2

The ordered pair is (2, 0).

Using the ordered pairs in Examples 2 through 4, create a pictorial representation of the linear equation in a Cartesian Coordinate System.

That is, let's use the following ordered pairs: (0, 6), (2, 0), (4, 6)

Example 6:

Graph the line by finding two points on the graph.

This is an equation with "ugly" numbers. Instead of working with fractions, let's be smart when selecting two values for x. That is, let's pick them in a way so that the values for y are integers.

Let's decide to pick 1 for x.

Then

The coordinates of the point are (1, 4).

Next, let's decide to pick 2 for x.

Then

The coordinates of the additional point are (2, 1).

Below is the graph of the line.

Example 7:

Graph the equations and in a Cartesian Coordinate System.

We know that the graphs of both equations are horizontal lines.  Notice that there is NO x-value. This means that x can take on any and all real numbers. However, ALL ordered pairs that are solutions to both equations have a value of y that is always and , respectively.

The easiest real number that x th can take on is 0. Therefore we know that the points and , respectively, lie on the graphs.

Example 8:

Graph the equations and in a Cartesian Coordinate System.

We know that the graphs of both equations are vertical lines.  Notice that there is NO y-value. This means that y can take on any and all real numbers. However, ALL ordered pairs that are solutions to both equations have a value of x that is always and , respectively.

The easiest real number that y th can take on is 0. Therefore we know that the points and , respectively, lie on the graphs.

Below are the graphs of the two equations in the same Cartesian Coordinate System.