# Graphs

## Use the Rectangular Coordinate System


### Plot Points on a Rectangular Coordinate System

Many maps, such as the Campus Map shown in , use a grid system to identify locations. Do you see the numbers  and  across the top and bottom of the map and the letters A, B, C, and D along the sides? Every location on the map can be identified by a number and a letter.
For example, the Student Center is in section 2B. It is located in the grid section above the number  and next to the letter B. In which grid section is the Stadium? The Stadium is in section 4D.
Just as maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. To create a rectangular coordinate system, start with a horizontal number line. Show both positive and negative numbers as you did before, using a convenient scale unit. This horizontal number line is called the .
Now, make a vertical number line passing through the  at  Put the positive numbers above  and the negative numbers below  See . This vertical line is called the .
Vertical grid lines pass through the integers marked on the  Horizontal grid lines pass through the integers marked on the  The resulting grid is the rectangular coordinate system.
The rectangular coordinate system is also called the  plane, the coordinate plane, or the Cartesian coordinate system (since it was developed by a mathematician named René Descartes.)
The  and the  form the rectangular coordinate system.  These axes divide a plane into four areas, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See .
In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point.
So how do the coordinates of a point help you locate a point on the  plane? 
Let’s try locating the point . In this ordered pair, the -coordinate is  and the -coordinate is .
We start by locating the  value,  on the  Then we lightly sketch a vertical line through  as shown in .
Now we locate the  value,  on the -axis and sketch a horizontal line through . The point where these two lines meet is the point with coordinates  We plot the point there, as shown in .
How do the signs affect the location of the points?
You may have noticed some patterns as you graphed the points in the two previous examples. 
For each point in Quadrant IV, what do you notice about the signs of the coordinates? 
What about the signs of the coordinates of the points in the third quadrant? The second quadrant? The first quadrant?
Can you tell just by looking at the coordinates in which quadrant the point (−2, 5) is located? In which quadrant is (2, −5) located?

We can summarize sign patterns of the quadrants as follows. Also see .
What if one coordinate is zero? Where is the point  located? Where is the point  located? The point   is on the y-axis and the point   is on the x-axis.

### Identify Points on a Graph

In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the  of a point on a graph, read the number on the  directly above or below the point. To identify the  of a point, read the number on the  directly to the left or right of the point. Remember, to write the ordered pair using the correct order 

### Verify Solutions to an Equation in Two Variables

All the equations we solved so far have been equations with one variable. In almost every case, when we solved the equation we got exactly one solution. The process of solving an equation ended with a statement such as  Then we checked the solution by substituting back into the equation.
Here’s an example of a linear equation in one variable, and its one solution.
But equations can have more than one variable. Equations with two variables can be written in the general form  An equation of this form is called a linear equation in two variables.
Notice that the word “line” is in linear. 
Here is an example of a linear equation in two variables,  and 
Is  a linear equation? It does not appear to be in the form  But we could rewrite it in this form.
By rewriting  as  we can see that it is a linear equation in two variables because it can be written in the form 
Linear equations in two variables have infinitely many solutions. For every number that is substituted for  there is a corresponding  value. This pair of values is a solution to the linear equation and is represented by the ordered pair  When we substitute these values of  and  into the equation, the result is a true statement because the value on the left side is equal to the value on the right side.

### Complete a Table of Solutions to a Linear Equation

In the previous examples, we substituted the  of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for  and then solve the equation for  Or, choose a value for  and then solve for 
We’ll start by looking at the solutions to the equation  we found in . We can summarize this information in a table of solutions.
To find a third solution, we’ll let  and solve for 
The ordered pair is a solution to .  We will add it to the table.

We can find more solutions to the equation by substituting any value of  or any value of  and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.

### Find Solutions to Linear Equations in Two Variables

To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either  or  We could choose  or any other value we want. But it’s a good idea to choose a number that’s easy to work with. We’ll usually choose  as one of our values.
We said that linear equations in two variables have infinitely many solutions, and we’ve just found one of them. Let’s find some other solutions to the equation 
Let’s find some solutions to another equation now.

### Key Concepts

1. Sign Patterns of the Quadrants

2. Coordinates of Zero

### Practice Makes Perfect

Plot Points on a Rectangular Coordinate System 
In the following exercises, plot each point on a coordinate grid.
In the following exercises, plot each point on a coordinate grid and identify the quadrant in which the point is located.
In the following exercises, plot each point on a coordinate grid.
Identify Points on a Graph 
In the following exercises, name the ordered pair of each point shown.
Verify Solutions to an Equation in Two Variables
In the following exercises, determine which ordered pairs are solutions to the given equation.
Find Solutions to Linear Equations in Two Variables 
In the following exercises, complete the table to find solutions to each linear equation.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.


ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.