11.1 Use the Rectangular Coordinate System

Learning Objectives

By the end of this section, you will be able to:

  • Plot points on a rectangular coordinate system
  • Identify points on a graph
  • Verify solutions to an equation in two variables
  • Complete a table of solutions to a linear equation
  • Find solutions to linear equations in two variables

Plot Points on a Rectangular Coordinate System

Many maps, such as the Campus Map shown in Figure 11.2, use a grid system to identify locations. Do you see the numbers 1,2,3,1,2,3, and 44 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 22 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 x-axis.

Now, make a vertical number line passing through the x-axisx-axis at 0.0. Put the positive numbers above 00 and the negative numbers below 0.0. See Figure 11.3. This vertical line is called the y-axis.

Vertical grid lines pass through the integers marked on the x-axis.x-axis. Horizontal grid lines pass through the integers marked on the y-axis.y-axis. The resulting grid is the rectangular coordinate system.

The rectangular coordinate system is also called the x-yx-y plane, the coordinate plane, or the Cartesian coordinate system (since it was developed by a mathematician named René Descartes.)

The x-axisx-axis and the y-axisy-axis 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 Figure 11.4.

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 x-yx-y plane?

Let’s try locating the point (2,5)(2,5). In this ordered pair, the xx-coordinate is 22 and the yy-coordinate is 55.

We start by locating the xx value, 2,2, on the x-axis.x-axis. Then we lightly sketch a vertical line through x=2,x=2, as shown in Figure 11.5.

Now we locate the yy value, 5,5, on the yy-axis and sketch a horizontal line through y=5y=5. The point where these two lines meet is the point with coordinates (2,5).(2,5). We plot the point there, as shown in Figure 11.6.

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?

What if one coordinate is zero? Where is the point (0,4)(0,4) located? Where is the point (−2,0)(−2,0) located? The point (0,4)(0,4) is on the y-axis and the point (−2,0)(−2,0) 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 x-coordinate of a point on a graph, read the number on the x-axis directly above or below the point. To identify the y-coordinate of a point, read the number on the y-axis directly to the left or right of the point. Remember, to write the ordered pair using the correct order (x,y).

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 x=4.x=4. 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.

By rewriting y=−5x+1y=−5x+1 as 5x+y=1,5x+y=1, we can see that it is a linear equation in two variables because it can be written in the form Ax+By=C.Ax+By=C.

Linear equations in two variables have infinitely many solutions. For every number that is substituted for x,x, there is a corresponding yy value. This pair of values is a solution to the linear equation and is represented by the ordered pair (x,y).(x,y). When we substitute these values of xx and yy 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 x- andy-valuesx- andy-values 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 xx and then solve the equation for y.y. Or, choose a value for yy and then solve for x.x.

We’ll start by looking at the solutions to the equation y=5x−1y=5x−1 we found in Example 11.9. We can summarize this information in a table of solutions.

We can find more solutions to the equation by substituting any value of xx or any value of yy 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 xx or y.y. We could choose 1,100,1,000,1,100,1,000, 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 00 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 3x+2y=6.

Let’s find some solutions to another equation now.

Section 11.1 Exercises

Practice Makes Perfect

Plot Points on a Rectangular Coordinate System

In the following exercises, plot each point on a coordinate grid.

Verify Solutions to an Equation in Two Variables

In the following exercises, determine which ordered pairs are solutions to the given equation.

Everyday Math

35Weight of a baby Mackenzie recorded her baby’s weight every two months. The baby’s age, in months, and weight, in pounds, are listed in the table, and shown as an ordered pair in the third column.

ⓐ Plot the points on a coordinate grid.