11.2 Use the Rectangular Coordinate System

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.

The figure shows a labeled grid representing the Campus Map. The columns are labeled 1 through 4 and the rows are labeled A through D. At position A-1 is the title Parking Garage. At position A-4 is a rectangle labeled Residence Halls. At position B-2 is a rectangle labeled Student Center. At position B-3 is a rectangle labeled Engineering Building. At position C-1 is a rectangle labeled Taylor Hall. At position C-2 is a rectangle labeled Library. At position C-4 is a rectangle labeled Tiger Field. At position D-4 is a rectangle labeled Stadium. — **Figure 11.2**

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.

The figure shows a number line with integer values labeled from -5 to 5.

Now, make a vertical number line passing through the $x -axis$ 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 $x -axis.$ Horizontal grid lines pass through the integers marked on the $y -axis.$ 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 graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. An arrow points to the horizontal axis with the label “x-axis”. An arrow points to the vertical axis with label “y-axis”. An arrow points to the intersection of the axes with label “origin”. — **Figure 11.3** The rectangular coordinate system.

The $x -axis$ and the $y -axis$ 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.

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled “I”, the top-left portion of the plane is labeled “II”, the bottom-left portion of the plane is labelled “III” and the bottom-right portion of the plane is labeled “IV” — **Figure 11.4** The four quadrants of the rectangular coordinate system

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 $open parenthesis 2, 5 close parenthesis$ . 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 $x -axis.$ Then we lightly sketch a vertical line through $x equals 2,$ as shown in Figure 11.5.

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There is a vertical dotted line passing through 2 on the x-axis. — **Figure 11.5**

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

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. An arrow starts at the origin and extends right to the number 2 on the x-axis. An arrow starts at the end of the first arrow at 2 on the x-axis and goes vertically 5 units to a point labeled “2, 5” in parentheses. — **Figure 11.6**

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?

A Cartesian coordinate system shows points (-2, 5) in Quadrant II and (2, -5) in Quadrant IV. The axes range from -7 to 7, with grid lines every unit. — **Figure 11.7**

We can summarize sign patterns of the quadrants as follows. Also see Figure 11.8.

**Table 11.1**
Quadrant I	Quadrant II	Quadrant III	Quadrant IV
(x,y)	(x,y)	(x,y)	(x,y)
(+,+)	(−,+)	(−,−)	(+,−)

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled “I” and “ordered pair +, +”, the top-left portion of the plane is labeled “II” and “ordered pair -, +”, the bottom-left portion of the plane is labelled “III” “ordered pair -, -” and the bottom-right portion of the plane is labeled “IV” and “ordered pair +, -”. — **Figure 11.8**

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

Example 6

Name the ordered pair of each point shown:

Point A is above $−3$ on the $x -axis,$ so the $x -coordinate$ of the point is $−3.$ The point is to the left of $3$ on the $y -axis,$ so the $y -coordinate$ of the point is $3 .$ The coordinates of the point are $open parenthesis −3, 3 close parenthesis.$

Point B is below $−1$ on the $x -axis,$ so the $x -coordinate$ of the point is $−1.$ The point is to the left of $−3$ on the $y -axis,$ so the $y -coordinate$ of the point is $−3.$ The coordinates of the point are $open parenthesis −1, −3 close parenthesis.$

Point C is above $2$ on the $x -axis,$ so the $x -coordinate$ of the point is $2 .$ The point is to the right of $4$ on the $y -axis,$ so the $y -coordinate$ of the point is $4 .$ The coordinates of the point are $open parenthesis 2, 4 close parenthesis.$

Point D is below $4$ on the $x - axis,$ so the $x -coordinate$ of the point is $4 .$ The point is to the right of $−4$ on the $y -axis,$ so the $y -coordinate$ of the point is $−4.$ The coordinates of the point are $open parenthesis 4, −4 close parenthesis.$

Example 7

Name the ordered pair of each point shown:

Point A is on the x-axis at $x = - 4$ .	The coordinates of point A are $open parenthesis minus 4, 0 close parenthesis$ .
Point B is on the y-axis at $y = - 2$	The coordinates of point B are $open parenthesis 0, minus 2 close parenthesis$ .
Point C is on the x-axis at $x equals 3$ .	The coordinates of point C are $open parenthesis 3, 0 close parenthesis$ .
Point D is on the y-axis at $y equals 1$ .	The coordinates of point D are $open parenthesis 0, 1 close parenthesis$ .

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 $x equals 4.$ 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.

$\begin{matrix} 3 x + 5 = 17 \\ 3 x = 12 \\ x = 4 \end{matrix}$

But equations can have more than one variable. Equations with two variables can be written in the general form $A x + B y = C .$ 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 $y :$

A series of equations is shown. The first line shows A x + B x = C. The “A” is red, the “B” is blue, and the “C” is turquoise. The second line shows x + 4 y = 8. The “4” is blue and the “8” is turquoise. The last line shows A =1 in red, B = 4 in blue, and C =8 in turquoise.

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


Add $5 x$ to both sides.
Simplify.
Use the Commutative Property to put it in $A x + B y = C .$

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

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 8

Determine which ordered pairs are solutions of the equation $x + 4 y = 8 :$

ⓐ $open parenthesis 0, 2 close parenthesis$
ⓑ $open parenthesis 2, −4 close parenthesis$
ⓒ $open parenthesis −4, 3 close parenthesis$

Substitute the $x - and y -values$ from each ordered pair into the equation and determine if the result is a true statement.

ⓐ $open parenthesis 0, 2 close parenthesis$	ⓑ $open parenthesis 2, −4 close parenthesis$	ⓒ $open parenthesis −4, 3 close parenthesis$

$open parenthesis 0, 2 close parenthesis$ is a solution.	$open parenthesis 2, −4 close parenthesis$ is not a solution.	$open parenthesis −4, 3 close parenthesis$ is a solution.

Example 9

Determine which ordered pairs are solutions of the equation. $y = 5 x - 1 :$

ⓐ $open parenthesis 0, −1 close parenthesis$
ⓑ $open parenthesis 1, 4 close parenthesis$
ⓒ $open parenthesis −2, −7 close parenthesis$

Substitute the $x -$ and $y -values$ from each ordered pair into the equation and determine if it results in a true statement.

ⓐ $open parenthesis 0, −1 close parenthesis$	ⓑ $open parenthesis 1, 4 close parenthesis$	ⓒ $open parenthesis −2, −7 close parenthesis$

$open parenthesis 0, −1 close parenthesis$ is a solution.	$open parenthesis 1, 4 close parenthesis$ is a solution.	$open parenthesis −2, −7 close parenthesis$ is not a solution.

Complete a Table of Solutions to a Linear Equation

In the previous examples, we substituted the $x - and y -values$ 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 $y = 5 x - 1$ we found in Example 9. We can summarize this information in a table of solutions.

$y = 5 x - 1$
$x$	$y$	$(x, y)$
$0$	$−1$	$open parenthesis 0, −1 close parenthesis$
$1$	$4$	$open parenthesis 1, 4 close parenthesis$

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

	$y = 5 x - 1$

Multiply.	$y = 10 - 1$
Simplify.	$y equals 9$

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

$y = 5 x - 1$
$x$	$y$	$(x, y)$
$0$	$−1$	$open parenthesis 0, −1 close parenthesis$
$1$	$4$	$open parenthesis 1, 4 close parenthesis$
$2$	$9$	$open parenthesis 2, 9 close parenthesis$

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.

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 12

Find a solution to the equation $3 x + 2 y = 6 .$

Step 1: Choose any value for one of the variables in the equation.		We can substitute any value we want for $x$ or any value for $y .$ Let's pick $x equals 0.$ What is the value of $y$ if $x equals 0$ ?
Step 2: Substitute that value into the equation. Solve for the other variable.	Substitute $0$ for $x .$ Simplify. Divide both sides by 2.
Step 3: Write the solution as an ordered pair.	So, when $x = 0, y = 3 .$	This solution is represented by the ordered pair $open parenthesis 0, 3 close parenthesis.$
Step 4: Check.	Is the result a true equation? Yes!

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 $3 x + 2 y = 6 .$

Example 13

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

To find solutions to $3 x + 2 y = 6,$ choose a value for $x$ or $y .$ Remember, we can choose any value we want for $x$ or $y .$ Here we chose $1$ for $x,$ and $0$ and $−3$ for $y .$

Substitute it into the equation.
Simplify. Solve.

Write the ordered pair.	$open parenthesis 2, 0 close parenthesis$	$(1, \frac{3}{2})$	$open parenthesis 4, −3 close parenthesis$

Check your answers.

$open parenthesis 2, 0 close parenthesis$	$(1, \frac{3}{2})$	$open parenthesis 4, −3 close parenthesis$

So $(2, 0), (1, \frac{3}{2})$ and $open parenthesis 4, −3 close parenthesis$ are all solutions to the equation $3 x + 2 y = 6 .$ In the previous example, we found that $open parenthesis 0, 3 close parenthesis$ is a solution, too. We can list these solutions in a table.

$3 x + 2 y = 6$
$x$	$y$	$(x, y)$
$0$	$3$	$open parenthesis 0, 3 close parenthesis$
$2$	$0$	$open parenthesis 2, 0 close parenthesis$
$1$	$\frac{3}{2}$	$(1, \frac{3}{2})$
$4$	$−3$	$open parenthesis 4, −3 close parenthesis$

Let’s find some solutions to another equation now.

Example 14

Find three solutions to the equation $x - 4 y = 8 .$


Choose a value for $x$ or $y .$
Substitute it into the equation.
Solve.
Write the ordered pair.	$open parenthesis 0, −2 close parenthesis$	$open parenthesis 8, 0 close parenthesis$	$open parenthesis 20, 3 close parenthesis$

So $open parenthesis 0, −2 close parenthesis, open parenthesis 8, 0 close parenthesis,$ and $open parenthesis 20, 3 close parenthesis$ are three solutions to the equation $x - 4 y = 8 .$

$x - 4 y = 8$
$x$	$y$	$(x, y)$
$0$	$−2$	$open parenthesis 0, −2 close parenthesis$
$8$	$0$	$open parenthesis 8, 0 close parenthesis$
$20$	$3$	$open parenthesis 20, 3 close parenthesis$

Remember, there are an infinite number of solutions to each linear equation. Any point you find is a solution if it makes the equation true.

Key Concepts

Sign Patterns of the Quadrants
Quadrant I Quadrant II Quadrant III Quadrant IV
(x,y) (x,y) (x,y) (x,y)
(+,+) (−,+) (−,−) (+,−)
Coordinates of Zero
- Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates ( a, 0).
- Points with a x-coordinate equal to 0 are on the y-axis, and have coordinates ( 0, b).
- The point (0, 0) is called the origin. It is the point where the x-axis and y-axis intersect.

Practice Makes Perfect

Plot Points on a Rectangular Coordinate System

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

$open parenthesis 3, 2 close parenthesis$

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 3, 2” is labeled

$open parenthesis 4, 1 close parenthesis$

$open parenthesis 1, 5 close parenthesis$

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 1, 5” is labeled

$open parenthesis 3, 4 close parenthesis$

$open parenthesis 4, 1 close parenthesis, open parenthesis 1, 4 close parenthesis$

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 1, 4” is labeled. The point “ordered pair 4, 1” is labeled.

$open parenthesis 3, 2 close parenthesis, open parenthesis 2, 3 close parenthesis$

$open parenthesis 3, 4 close parenthesis, open parenthesis 4, 3 close parenthesis$

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 3, 4” is labeled. The point “ordered pair 4, 3” is labeled.

In the following exercises, plot each point on a coordinate grid and identify the quadrant in which the point is located.

ⓐ $open parenthesis −4, 2 close parenthesis$
ⓑ $open parenthesis −1, −2 close parenthesis$
ⓒ $open parenthesis 3, −5 close parenthesis$
ⓓ $(2, \frac{5}{2})$

ⓐ $open parenthesis −2, −3 close parenthesis$
ⓑ $open parenthesis 3, −3 close parenthesis$
ⓒ $open parenthesis −4, 1 close parenthesis$
ⓓ $(1, \frac{3}{2})$

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The quadrants are labeled I, II, III, and IV. The point (-1, 1) is labeled a, the point (-2, -1) is labeled b. The point (1, -4) is labeled c, and the point (3, 7/2) is labeled d.

ⓐ $open parenthesis −1, 1 close parenthesis$
ⓑ $open parenthesis −2, −1 close parenthesis$
ⓒ $open parenthesis 1, −4 close parenthesis$
ⓓ $(3, \frac{7}{2})$

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

ⓐ $open parenthesis 3, −2 close parenthesis$
ⓑ $open parenthesis −3, 2 close parenthesis$
ⓒ $open parenthesis −3, −2 close parenthesis$
ⓓ $open parenthesis 3, 2 close parenthesis$

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point (3, -2) is labeled a, the point (-3, 2) is labeled b. The point (-3, -2) is labeled c, and the point (3, 2) is labeled d.

ⓐ $open parenthesis 4, −1 close parenthesis$
ⓑ $open parenthesis −4, 1 close parenthesis$
ⓒ $open parenthesis −4, −1 close parenthesis$
ⓓ $open parenthesis 4, 1 close parenthesis$

ⓐ $open parenthesis −2, 0 close parenthesis$
ⓑ $open parenthesis −3, 0 close parenthesis$
ⓒ $open parenthesis 0, 4 close parenthesis$
ⓓ $open parenthesis 0, 2 close parenthesis$

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point (-2, 0) is labeled a, the point (-3, 0) is labeled b. The point (0, 4) is labeled c, and the point (0, 2) is labeled d.

Identify Points on a Graph

In the following exercises, name the ordered pair of each point shown.

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -4, 1” is labeled “A”. The point “ordered pair -3, -4” is labeled “B”.

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 4, 3” is labeled “D”. The point “ordered pair 1, -3” is labeled “C”.

C(1, -3) D(4, 3)

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -3, -2” is labeled “X”. The point “ordered pair 5, -1” is labeled “Y”.

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -2, 4” is labeled “S”. The point “ordered pair -4, -2” is labeled “T”.

S(-2, 4) T(-4, -2)

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -2, 0” is labeled “B”. The point “ordered pair 0, -2” is labeled “A”.

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -1, 0” is labeled “D”. The point “ordered pair 0, -1” is labeled “C”.

C(0, -1) D(-1, 0)

The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 3, 0” is labeled “T”. The point “ordered pair -4, 0” is labeled “S”.

Verify Solutions to an Equation in Two Variables

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

$2 x + y = 6$

ⓐ $open parenthesis 1, 4 close parenthesis$
ⓑ $open parenthesis 3, 0 close parenthesis$

ⓐ, ⓑ

$x + 3 y = 9$

ⓐ $open parenthesis 0, 3 close parenthesis$
ⓑ $open parenthesis 6, 1 close parenthesis$

$4 x - 2 y = 8$

ⓐ $open parenthesis 3, 2 close parenthesis$
ⓑ $open parenthesis 1, 4 close parenthesis$

ⓐ, ⓒ

$3 x - 2 y = 12$

ⓐ $open parenthesis 4, 0 close parenthesis$
ⓑ $open parenthesis 2, −3 close parenthesis$

$y = 4 x + 3$

ⓐ $open parenthesis 4, 3 close parenthesis$
ⓑ $open parenthesis −1, −1 close parenthesis$

ⓑ, ⓒ

$y = 2 x - 5$

ⓐ $open parenthesis 0, −5 close parenthesis$
ⓑ $open parenthesis 2, 1 close parenthesis$

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

ⓐ $open parenthesis 2, 0 close parenthesis$
ⓑ $open parenthesis −6, −4 close parenthesis$

ⓐ, ⓑ

$y = \frac{1}{3} x + 1$

ⓐ $open parenthesis −3, 0 close parenthesis$
ⓑ $open parenthesis 9, 4 close parenthesis$

Find Solutions to Linear Equations in Two Variables

In the following exercises, complete the table to find solutions to each linear equation.

$y = 2 x - 4$

$x$	$y$	$(x, y)$
$−1$
$0$
$2$

$x$	$y$	$(x, y)$
$−1$	$−6$	$open parenthesis −1, −6 close parenthesis$
$0$	$−4$	$open parenthesis 0, −4 close parenthesis$
$2$	$0$	$open parenthesis 2, 0 close parenthesis$

$y = 3 x - 1$

$x$	$y$	$(x, y)$
$−1$
$0$
$2$

$y = - x + 5$

$x$	$y$	$(x, y)$
$−2$
$0$
$3$

$x$	$y$	$(x, y)$
$−2$	$7$	$open parenthesis −2, 7 close parenthesis$
$0$	$5$	$open parenthesis 0, 5 close parenthesis$
$3$	$2$	$open parenthesis 3, 2 close parenthesis$

$y = \frac{1}{3} x + 1$

$x$	$y$	$(x, y)$
$0$
$3$
$6$

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

$x$	$y$	$(x, y)$
$−2$
$0$
$2$

$x$	$y$	$(x, y)$
$−2$	$1$	$open parenthesis −2, 1 close parenthesis$
$0$	$−2$	$open parenthesis 0, −2 close parenthesis$
$2$	$−5$	$open parenthesis 2, −5 close parenthesis$

$x + 2 y = 8$

$x$	$y$	$(x, y)$
$0$
$4$
	$0$

Everyday Math

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.

$Age$	$Weight$	$(x, y)$
$0$	$7$	$open parenthesis 0, 7 close parenthesis$
$2$	$11$	$open parenthesis 2, 11 close parenthesis$
$4$	$15$	$open parenthesis 4, 15 close parenthesis$
$6$	$16$	$open parenthesis 6, 16 close parenthesis$
$8$	$19$	$open parenthesis 8, 19 close parenthesis$
$10$	$20$	$open parenthesis 10, 20 close parenthesis$
$12$	$21$	$open parenthesis 12, 21 close parenthesis$

ⓑ Why is only Quadrant I needed?

ⓐ
ⓑ Age and weight are only positive.

Weight of a child Latresha recorded her son’s height and weight every year. His height, in inches, 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.

$\begin{matrix} Height \\ x \end{matrix}$	$\begin{matrix} Weight \\ y \end{matrix}$	$\begin{matrix} (x, y) \end{matrix}$
$28$	$22$	$open parenthesis 28, 22 close parenthesis$
$31$	$27$	$open parenthesis 31, 27 close parenthesis$
$33$	$33$	$open parenthesis 33, 33 close parenthesis$
$37$	$35$	$open parenthesis 37, 35 close parenthesis$
$40$	$41$	$open parenthesis 40, 41 close parenthesis$
$42$	$45$	$open parenthesis 42, 45 close parenthesis$

ⓑ Why is only Quadrant I needed?

Writing Exercises

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

Answers may vary.

How do you determine if an ordered pair is a solution to a given equation?

Self Check

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

An empty self-assessment checklist for students to rate their confidence in math skills like plotting points, identifying points on graphs, and solving linear equations with categories: Confidently, With some help, No-I don't get it! — **Figure 11.9**

ⓑ 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.


$y = 4 x - 2$	$y = 4 x - 2$	$y = 4 x - 2$

$y = 0 - 2$	$y = −4 - 2$	$y = 8 - 2$
$y equals −2$	$y equals −6$	$y equals 6$
$open parenthesis 0, −2 close parenthesis$	$open parenthesis −1, −6 close parenthesis$	$open parenthesis 2, 6 close parenthesis$

$y = 6 x + 1$
$x$	$y$	$(x, y)$
$0$	$1$	$open parenthesis 0, 1 close parenthesis$
$1$	$7$	$open parenthesis 1, 7 close parenthesis$
$−2$	$−11$	$open parenthesis −2, −11 close parenthesis$

$5 x - 4 y = 20$
$x$	$y$	$(x, y)$
$0$	$−5$	$open parenthesis 0, −5 close parenthesis$
$4$	$0$	$open parenthesis 4, 0 close parenthesis$
$8$	$5$	$open parenthesis 8, 5 close parenthesis$

$3 x - 4 y = 12$
$x$	$y$	$(x, y)$
$0$	$−3$	$open parenthesis 0, −3 close parenthesis$
$4$	$0$	$open parenthesis 4, 0 close parenthesis$
$−4$	$−6$	$open parenthesis −4, −6 close parenthesis$