11.1 Use the Rectangular Coordinate System

Learning objectives

Learning objectives

• 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 $1,2,3,$ and $4$ 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 $2$ and next to the letter B. In which grid section is the Stadium? The Stadium is in section 4D.

Example 11.1

Solution

1. Read the grid naming rule. A grid section combines the number printed along the bottom with the letter printed along the side, so each location requires one number and one letter.
2. Locate the Residence Halls. The Residence Halls align with the number $4$ along the bottom and the letter $A$ along the side, placing them in grid section $4A$.
3. Locate Tiger Field. Tiger Field aligns with the number $4$ and the letter $C$, so it is the landmark in grid section $4C$.
4. Compare the two readings. Both landmarks use the same bottom number, but their different side letters place them in distinct grid sections.

Takeaway: Read a grid section from the number along the bottom and the letter along the side: the Residence Halls are in 4A and Tiger Field occupies 4C.

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 $x\mathrm{-axis}$ at $0.$ Put the positive numbers above $0$ and the negative numbers below $0.$ See Figure 11.3. This vertical line is called the y-axis.

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

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

The $x\mathrm{-axis}$ and the $y\mathrm{-axis}$ 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 $x-y$ plane?

Let’s try locating the point $(2,5)$. In this ordered pair, the $x$-coordinate is $2$ and the $y$-coordinate is $5$.

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

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

Example 11.2

Solution

1. Interpret the first coordinate. In $(1,3)$, the $x$-coordinate is $1$, so move one unit right from the origin to $x=1$.
2. Interpret the second coordinate. The $y$-coordinate is $3$, so move three units up from $x=1$ and mark the intersection as $(1,3)$.
3. Plot the reversed pair. For $(3,1)$, move three units right to $x=3$ and one unit up to $y=1$, marking $(3,1)$.
4. Compare the locations. The two points use the same coordinate values in opposite orders and lie in different quadrants, so reversing an ordered pair changes its location.

Takeaway: The order of the pair matters: $(1,3)$ and $(3,1)$ use the same digits but place them in different quadrants because $x$ comes first and $y$ second.

Example 11.3

Solution

1. Use the sign pattern. A negative $x$-coordinate places a point left of the $y$-axis, and a negative $y$-coordinate places it below the $x$-axis.
2. Locate $(-1,3)$. Its $x$-coordinate is negative and its $y$-coordinate is positive, so the point is in Quadrant II.
3. Locate $(-3,-4)$. Both coordinates are negative, so the point is left and below the origin in Quadrant III.
4. Locate $(2,-3)$. Its $x$-coordinate is positive and its $y$-coordinate is negative, so the point is in Quadrant IV.
5. Locate $(3,\frac{5}{2})$. Since $\frac{5}{2}=2.5$ is positive, the point lies in Quadrant I, halfway between $2$ and $3$ units above the $x$-axis.
6. State the quadrant rule. The signs of the coordinates determine each quadrant, while the coordinate magnitudes determine the point's distance from each axis.

Takeaway: The signs of the coordinates name the quadrant: $(-1,3)$ lies in Quadrant II, $(-3,-4)$ in Quadrant III, and $(2,-3)$ in Quadrant IV.

How do the signs affect the location of the points?

Example 11.4

Solution

1. Locate the horizontal coordinate. For $(-5,2)$, move to $x=-5$ on the $x$-axis; the point is $2$ units above the axis, so it lies in Quadrant II.
2. Locate the remaining points. The point $(-5,-2)$ is below the $x$-axis in Quadrant III, $(5,2)$ is above the $x$-axis in Quadrant I, and $(5,-2)$ is below it in Quadrant IV.
3. Check the sign patterns. The four ordered pairs produce sign patterns $(-,+)$, $(-,-)$, $(+,+)$, and $(+,-)$, respectively, matching the quadrant diagram and the examples $(-2,5)$ and $(2,-5)$ shown in the figures.
4. State the plotted locations. Thus the points are $(-5,2)$ in Quadrant II, $(-5,-2)$ in Quadrant III, $(5,2)$ in Quadrant I, and $(5,-2)$ in Quadrant IV.

Takeaway: The signs of the coordinates determine the quadrant, with the $x$-coordinate controlling left or right and the $y$-coordinate controlling up or down.

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 Figure 11.7.

Table

11.1

What if one coordinate is zero? Where is the point $(0,4)$ located? Where is the point $(-2,0)$ located? The point $(0,4)$ is on the y-axis and the point $(-2,0)$ is on the x-axis.

Example 11.5

Solution

1. Apply the axis rule. A point with $x=0$ lies on the $y$-axis, a point with $y=0$ lies on the $x$-axis, and two zero coordinates give the origin.
2. Plot $(0,5)$. Since its $x$-coordinate is zero, start on the $y$-axis and move up $5$ units to plot the point.
3. Plot $(4,0)$. Since its $y$-coordinate is zero, move $4$ units right on the $x$-axis to plot the point.
4. Plot $(-3,0)$. Its $y$-coordinate is zero, so it lies on the $x$-axis $3$ units to the left of the origin.
5. Plot $(0,0)$. Both coordinates are zero, so this point is the origin where the two axes intersect.
6. Plot $(0,-1)$. Its $x$-coordinate is zero, so it lies on the $y$-axis $1$ unit below the origin.

Takeaway: A zero coordinate puts the point on an axis: $(0,5)$ is on the $y$-axis, $(4,0)$ and $(-3,0)$ are on the $x$-axis, and $(0,0)$ is the origin.

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 $(x,y).$

Example 11.6

Solution

1. Read point A. Projecting A to the axes gives $x=-3$ and $y=3$, so its ordered pair is $(-3,3)$.
2. Read point B. Projecting B to the axes gives $x=-1$ and $y=-3$, so its ordered pair is $(-1,-3)$.
3. Read point C. Projecting C to the axes gives $x=2$ and $y=4$, so its ordered pair is $(2,4)$.
4. Read point D. Projecting D to the axes gives $x=4$ and $y=-4$, so its ordered pair is $(4,-4)$.
5. Check the coordinate order. Each point's horizontal value is written first and its vertical value second, and the negative directions match the positions left of or below the origin.

Takeaway: Read the $x$-coordinate across the horizontal axis and the $y$-coordinate up the vertical axis, which for point A gives $(-3,3)$.

Example 11.7

Solution

1. Apply the axis rule. A point on the $x$-axis has $y=0$, while a point on the $y$-axis has $x=0$, so the zero coordinate identifies the axis.
2. Read point A. Point A lies on the $x$-axis at $x=-4$, so its ordered pair is $(-4,0)$.
3. Read point B. Point B lies on the $y$-axis at $y=-2$, so its ordered pair is $(0,-2)$.
4. Read point C. Point C lies on the $x$-axis at $x=3$, so its ordered pair is $(3,0)$.
5. Read point D. Point D lies on the $y$-axis at $y=1$, so its ordered pair is $(0,1)$.

Takeaway: A point on an axis has one zero coordinate: A is $(-4,0)$, B is $(0,-2)$, C is $(3,0)$ and D is $(0,1)$.

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 $x=4.$ 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 $Ax+By=C.$ 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, $x$ and $y:$

Is $y=-5x+1$ a linear equation? It does not appear to be in the form $Ax+By=C.$ But we could rewrite it in this form.

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

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

Example 11.8

Solution

1. Test the first ordered pair. Substituting $(0,2)$ into $x+4y=8$ gives $0+4(2)=8$, so $8=8$ and $(0,2)$ is a solution.
2. Test the second ordered pair. Substituting $(2,-4)$ gives $2+4(-4)=2-16=-14$, which is not equal to $8$, so $(2,-4)$ is not a solution.
3. Test the third ordered pair. Substituting $(-4,3)$ gives $-4+4(3)=-4+12=8$, so $(0,2)$ and $(-4,3)$ are solutions.
4. Interpret the verification figures. The figures show the same three substitutions: the checks end in $8=8$ for $(0,2)$ and $(-4,3)$, while the check for $(2,-4)$ ends in $-14\ne8$.

Takeaway: An ordered pair is a solution only when substitution produces a true equality between the two sides of the equation.

Example 11.9

Solution

1. Evaluate the first ordered pair. For $(0,-1)$, the rule gives $5(0)-1=-1$, which equals the proposed $y$-value, so $(0,-1)$ is a solution.
2. Evaluate the second ordered pair. For $(1,4)$, the rule gives $5(1)-1=4$, which equals the proposed $y$-value, so $(1,4)$ is a solution.
3. Evaluate the third ordered pair. For $(-2,-7)$, the rule gives $5(-2)-1=-10-1=-11$, but $-11\ne-7$, so $(-2,-7)$ is not a solution.
4. Confirm with the verification figures. The figures display the same computations, ending in $-1=-1$ and $4=4$ with checks, and in $-7\ne-11$ for the third pair.

Takeaway: To test a pair in an equation solved for $y$, calculate $y$ from the first coordinate and compare that result with the second coordinate.

Complete a Table of Solutions to a Linear Equation

In the previous examples, we substituted the $x\mathrm{- and}\,y\mathrm{-values}$ 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 $x$ and then solve the equation for $y.$ Or, choose a value for $y$ and then solve for $x.$

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

To find a third solution, we’ll let $x=2$ and solve for $y.$

The ordered pair is a solution to $y=5x-1$. We will add it to the table.

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

Example 11.10

Solution

1. Substitute $x=0$. The rule gives $y=4(0)-2=-2$, so the first ordered pair is $(0,-2)$.
2. Substitute $x=-1$. The rule gives $y=4(-1)-2=-4-2=-6$, so the second ordered pair is $(-1,-6)$.
3. Substitute $x=2$. The rule gives $y=4(2)-2=8-2=6$, so the third ordered pair is $(2,6)$.
4. Complete the table. Record $y=-2,-6,$ and $6$ beside the corresponding $x$-values, then write the three ordered pairs in $x,y$ order.

Takeaway: Substituting $x=0$, $x=-1$ and $x=2$ into $y=4x-2$ gives the solutions $(0,-2)$, $(-1,-6)$ and $(2,6)$ — each chosen $x$ fixes exactly one $y$.

Example 11.11

Solution

1. Use $x=0$. Substitution into $5x-4y=20$ gives $-4y=20$, so $y=-5$ and the ordered pair is $(0,-5)$.
2. Use $y=0$. Substitution gives $5x=20$, so $x=4$ and the ordered pair is $(4,0)$.
3. Use $y=5$. Substitution gives $5x-4(5)=20$, hence $5x-20=20$ and $5x=40$, so $x=8$ and the pair is $(8,5)$.
4. Complete the table. Record these three coordinate pairs in $x,y$ order, and each satisfies the original equation because both sides equal $20$.

Takeaway: Choosing the easy values $x=0$, $y=0$ and $y=5$ in $5x-4y=20$ produces $(0,-5)$, $(4,0)$ and $(8,5)$.

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 $x$ or $y.$ We could choose $1,100,1,000,$ 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 $0$ as one of our values.

Example 11.12

Solution

1. Choose a zero coordinate. Set $x=0$ in $3x+2y=6$ to find the $y$-intercept and obtain $0+2y=6$.
2. Solve for the second coordinate. Dividing by $2$ gives $y=3$, so the ordered pair is $(0,3)$.
3. Verify the solution. Substitution gives $3(0)+2(3)=0+6=6$, so $(0,3)$ satisfies the equation.
4. Read the verification figures. The figures show the sequence $x=0$, $3(0)+2y=6$, and the confirmed equality $6=6$ for the point $(0,3)$.

Takeaway: Setting one variable to zero is a reliable way to find an intercept because it produces an equation with only one variable.

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 $3x+2y=6.$

Example 11.13

Solution

1. Choose a value for $x$. Solve the equation for $y$ first: subtracting $3x$ gives $2y=6-3x$, so $y=\frac{6-3x}{2}$ for any value of $x$ we pick.
2. Substitute $x=2$. The equation becomes $3(2)+2y=6$, so $2y=0$ and $y=0$; the ordered pair $(2,0)$ is a solution.
3. Substitute $x=1$. The equation becomes $3(1)+2y=6$, so $2y=3$ and $y=\frac{3}{2}$; the ordered pair $\left(1,\frac{3}{2}\right)$ is a solution.
4. Substitute $x=4$. The equation becomes $3(4)+2y=6$, so $2y=-6$ and $y=-3$; the ordered pair $(4,-3)$ is a solution.
5. Check each pair. Substituting gives $6+0=6$, $3+3=6$, and $12-6=6$, so $(2,0)$, $\left(1,\frac{3}{2}\right)$, and $(4,-3)$ all satisfy $3x+2y=6$.

Takeaway: Any value may be substituted into $3x+2y=6$: $x=2$ gives $(2,0)$, $y=3$ gives $(0,3)$, and $y=-3$ gives $(4,-3)$.

Let’s find some solutions to another equation now.

Example 11.14

Solution

1. Choose the form to substitute into. Solving $x-4y=8$ for $x$ gives $x=8+4y$, so a value chosen for $y$ determines $x$ immediately.
2. Let $y=0$ to find the $x$-intercept. Substituting gives $x-4(0)=8$, so $x=8$ and the ordered pair $(8,0)$ is a solution.
3. Let $x=0$ to find the $y$-intercept. Substituting gives $0-4y=8$, so $y=-2$ and the ordered pair $(0,-2)$ is a solution.
4. Let $y=3$ to find a third point. Substituting gives $x-4(3)=8$, so $x=8+12=20$ and the ordered pair $(20,3)$ is a solution.
5. Check each pair. Substituting gives $8-0=8$, $0-(-8)=8$, and $20-12=8$, so $(8,0)$, $(0,-2)$, and $(20,3)$ all satisfy $x-4y=8$.

Takeaway: $(0,-2)$, $(8,0)$ and $(20,3)$ all satisfy $x-4y=8$; a linear equation has infinitely many solutions, so any point that makes it true is one of them.

Section 11.1 Exercises

Section 11.1 Exercises

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.

1. Name the ordered pair of each point shown:

2. Name the ordered pair of each point shown:

1. Name the ordered pair of each point shown:

2. Name the ordered pair of each point shown:

1. Use the map in Figure 11.2.

1. ⓐ Find the grid section of Taylor Hall.

2. ⓑ What is located in section 3B?

2. Use the map in Figure 11.2.

1. ⓐ Find the grid section of the Parking Garage.

2. ⓑ What is located in section 2C?

1. Plot each point on a rectangular coordinate system and identify the quadrant in which the point is located.

1. ⓐ $\,(-2,1)\,$

2. ⓑ $\,(-3,-1)$

3. ⓒ $\,(4,-4)$

4. ⓓ $\,(-4,\frac{3}{2})$

2. Plot each point on a rectangular coordinate system and identify the quadrant in which the point is located.

1. ⓐ $\,(-4,1)$

2. ⓑ $\,(-2,3)$

3. ⓒ $\,(2,-5)$

4. ⓓ $\,(-3,\frac{5}{2})$

1. Determine which ordered pairs are solutions to the given equation: $2x+3y=6$

1. ⓐ $\,(3,0)$

2. ⓑ $\,(2,0)$

3. ⓒ $\,(6,-2)$

2. Determine which ordered pairs are solutions to the given equation: $4x-y=8$

1. ⓐ $\,(0,8)$

2. ⓑ $\,(2,0)$

3. ⓒ $\,(1,-4)$

1. Determine which ordered pairs are solutions of the given equation: $y=4x-3$

1. ⓐ $\,(0,3)$

2. ⓑ $\,(1,1)$

3. ⓒ $\,(1,0)$

2. Determine which ordered pairs are solutions of the given equation: $y=-2x+6$

1. ⓐ $\,(0,6)$

2. ⓑ $\,(1,4)$

3. ⓒ $\,(-2,-2)$

1. Complete the table to find three solutions to the equation: $y=3x-1.$

2. Complete the table to find three solutions to the equation: $y=6x+1$

1. Complete the table to find three solutions to the equation: $2x-5y=20.$

2. Complete the table to find three solutions to the equation: $3x-4y=12.$

1. Find a solution to the equation: $4x+3y=12.$

2. Find a solution to the equation: $2x+4y=8.$

1. Find three solutions to the equation: $2x+3y=6.$

2. Find three solutions to the equation: $4x+2y=8.$

1. Find three solutions to the equation: $4x+y=8.$

2. Find three solutions to the equation: $x+5y=10.$

1. $2x+y=6$

1. ⓐ $\,(1,4)$

2. ⓑ $\,(3,0)$

3. ⓒ $\,(2,3)$

2. $4x-2y=8$

1. ⓐ $\,(3,2)$

2. ⓑ $\,(1,4)$

3. ⓒ $\,(0,-4)$

1. $y=4x+3$

1. ⓐ $\,(4,3)$

2. ⓑ $\,(-1,-1)$

3. ⓒ $\,(\frac{1}{2},5)$

2. $y=\frac{1}{2}x-1$

1. ⓐ $\,(2,0)$

2. ⓑ $\,(-6,-4)$

3. ⓒ $\,(-4,-1)$

1. $y=2x-4$

2. $y=-x+5$

1. $y=-\frac{3}{2}x-2$

2. Weight 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.

ⓑ Why is only Quadrant I needed?

1. Have you ever used a map with a rectangular coordinate system? Describe the map and how you used it.