# 11.1 Use the rectangular coordinate system

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

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

Let’s find some solutions to another equation now.

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

## Solution

 Choose a value for $x$ or $y.$ Substitute it into the equation. Solve. Write the ordered pair. $\left(0,-2\right)$ $\left(8,0\right)$ $\left(20,3\right)$

So $\left(0,-2\right),\left(8,0\right),$ and $\left(20,3\right)$ are three solutions to the equation $x-4y=8.$

$x-4y=8$
$x$ $y$ $\left(x,y\right)$
$0$ $-2$ $\left(0,-2\right)$
$8$ $0$ $\left(8,0\right)$
$20$ $3$ $\left(20,3\right)$

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.

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

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

## Key concepts

• Sign Patterns of the Quadrants
( 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.

$\left(3,2\right)$

$\left(4,1\right)$

$\left(1,5\right)$

$\left(3,4\right)$

$\left(4,1\right),\left(1,4\right)$

$\left(3,2\right),\left(2,3\right)$

$\left(3,4\right),\left(4,3\right)$

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

1. $\phantom{\rule{0.2em}{0ex}}\left(-4,2\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(-1,-2\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(3,-5\right)$
4. $\phantom{\rule{0.2em}{0ex}}\left(2,\frac{5}{2}\right)$

1. $\phantom{\rule{0.2em}{0ex}}\left(-2,-3\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(3,-3\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(-4,1\right)$
4. $\phantom{\rule{0.2em}{0ex}}\left(1,\frac{3}{2}\right)$

1. $\phantom{\rule{0.2em}{0ex}}\left(-1,1\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(-2,-1\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(1,-4\right)$
4. $\phantom{\rule{0.2em}{0ex}}\left(3,\frac{7}{2}\right)$

1. $\phantom{\rule{0.2em}{0ex}}\left(3,-2\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(-3,2\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(-3,-2\right)$
4. $\phantom{\rule{0.2em}{0ex}}\left(3,2\right)$

1. $\phantom{\rule{0.2em}{0ex}}\left(4,-1\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(-4,1\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(-4,-1\right)$
4. $\phantom{\rule{0.2em}{0ex}}\left(4,1\right)$

1. $\phantom{\rule{0.2em}{0ex}}\left(-2,0\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(-3,0\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(0,4\right)$
4. $\phantom{\rule{0.2em}{0ex}}\left(0,2\right)$

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.

$2x+y=6$

1. $\phantom{\rule{0.2em}{0ex}}\left(1,4\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(3,0\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(2,3\right)$

,

$x+3y=9$

1. $\phantom{\rule{0.2em}{0ex}}\left(0,3\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(6,1\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(-3,-3\right)$

$4x-2y=8$

1. $\phantom{\rule{0.2em}{0ex}}\left(3,2\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(1,4\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(0,-4\right)$

,

$3x-2y=12$

1. $\phantom{\rule{0.2em}{0ex}}\left(4,0\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(2,-3\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(1,6\right)$

$y=4x+3$

1. $\phantom{\rule{0.2em}{0ex}}\left(4,3\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(-1,-1\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(\frac{1}{2},5\right)$

,

$y=2x-5$

1. $\phantom{\rule{0.2em}{0ex}}\left(0,-5\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(2,1\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(\frac{1}{2},-4\right)$

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

1. $\phantom{\rule{0.2em}{0ex}}\left(2,0\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(-6,-4\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(-4,-1\right)$

,

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

1. $\phantom{\rule{0.2em}{0ex}}\left(-3,0\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(9,4\right)$
3. $\phantom{\rule{0.2em}{0ex}}\left(-6,-1\right)$

Find Solutions to Linear Equations in Two Variables

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

$y=2x-4$

$x$ $y$ $\left(x,y\right)$
$-1$
$0$
$2$
$x$ $y$ $\left(x,y\right)$
$-1$ $-6$ $\left(-1,-6\right)$
$0$ $-4$ $\left(0,-4\right)$
$2$ $0$ $\left(2,0\right)$

$y=3x-1$

$x$ $y$ $\left(x,y\right)$
$-1$
$0$
$2$

$y=-x+5$

$x$ $y$ $\left(x,y\right)$
$-2$
$0$
$3$
$x$ $y$ $\left(x,y\right)$
$-2$ $7$ $\left(-2,7\right)$
$0$ $5$ $\left(0,5\right)$
$3$ $2$ $\left(3,2\right)$

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

$x$ $y$ $\left(x,y\right)$
$0$
$3$
$6$

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

$x$ $y$ $\left(x,y\right)$
$-2$
$0$
$2$
$x$ $y$ $\left(x,y\right)$
$-2$ $1$ $\left(-2,1\right)$
$0$ $-2$ $\left(0,-2\right)$
$2$ $-5$ $\left(2,-5\right)$

$x+2y=8$

$x$ $y$ $\left(x,y\right)$
$0$
$4$
$0$

## 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.

 $\text{Age}$ $\text{Weight}$ $\left(x,y\right)$ $0$ $7$ $\left(0,7\right)$ $2$ $11$ $\left(2,11\right)$ $4$ $15$ $\left(4,15\right)$ $6$ $16$ $\left(6,16\right)$ $8$ $19$ $\left(8,19\right)$ $10$ $20$ $\left(10,20\right)$ $12$ $21$ $\left(12,21\right)$

Why is only Quadrant I needed?

1. 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.

 $\begin{array}{c}\text{Height}\hfill \\ x\hfill \end{array}$ $\begin{array}{c}\text{Weight}\hfill \\ y\hfill \end{array}$ $\begin{array}{}\\ \left(x,y\right)\hfill \end{array}$ $28$ $22$ $\left(28,22\right)$ $31$ $27$ $\left(31,27\right)$ $33$ $33$ $\left(33,33\right)$ $37$ $35$ $\left(37,35\right)$ $40$ $41$ $\left(40,41\right)$ $42$ $45$ $\left(42,45\right)$

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.

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.

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. Who 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.

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