Introduction to Equations in Two Variables; Cartesian Coordinate System

INTRODUCTION TO EQUATIONS IN TWO VARIABLES; CARTESIAN COORDINATE SYSTEM

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

What is discussed in this lecture?

1. Introduction to Equations in Two Variables
2. The Cartesian Coordinate System

1. Introduction to Equations in Two Variables

Up to this point, we discusses equations in one variable. For example, all of the linear equations we solved so far 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 equations in two variable.

Linear Equations: or . We know that they are linear equations because both variables are raised to the power of 1.

Quadratic Equations: or . We know that they are quadratic equations because one variable is raised to the power of 2.

There are infinitely many other equations in two variable. The only ones we are going to discuss in detail in this course are linear equations in two variables.

2. The Cartesian Coordinate System

Often we want to show a pictorial representation or graphs 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. 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 System.