# 4.1 Use the rectangular coordinate system  (Page 5/12)

 Page 5 / 12

Find three solutions to this equation: $y=-2x+3$ .

Find three solutions to this equation: $y=-4x+1$ .

We have seen how using zero as one value of $x$ makes finding the value of $y$ easy. When an equation is in standard form, with both the $x$ and $y$ on the same side of the equation, it is usually easier to first find one solution when $x=0$ find a second solution when $y=0$ , and then find a third solution.

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

## Solution

We can substitute any value we want for $x$ or any value for $y$ . Since the equation is in standard form, let’s pick first $x=0$ , then $y=0$ , and then find a third point.      Substitute the value into the equation.   Simplify.   Solve.      Write the ordered pair. (0, 3) (2, 0) $\left(1,\frac{3}{2}\right)$ Check. $3x+2y=6\phantom{\rule{1.3em}{0ex}}$ $3x+2y=6\phantom{\rule{1.3em}{0ex}}$ $3x+2y=6\phantom{\rule{1.3em}{0ex}}$ $3\cdot 0+2\cdot 3\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$ $3\cdot 2+2\cdot 0\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$ $3\cdot 1+2\cdot \frac{3}{2}\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$ $0+6\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$ $6+0\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$ $3+3\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$ $6=6✓$ $6=6✓$ $6=6✓$

So $\left(0,3\right)$ , $\left(2,0\right)$ , and $\left(1,\frac{3}{2}\right)$ are all solutions to the equation $3x+2y=6$ . We can list these three solutions in [link] .

 $3x+2y=6$ $x$ $y$ $\left(x,y\right)$ 0 3 $\left(0,3\right)$ 2 0 $\left(2,0\right)$ 1 $\frac{3}{2}$ $\left(1,\frac{3}{2}\right)$

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

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

## Key concepts

• Sign Patterns of the Quadrants
$\begin{array}{cccccccccc}\text{Quadrant I}\hfill & & & \text{Quadrant II}\hfill & & & \text{Quadrant III}\hfill & & & \text{Quadrant IV}\hfill \\ \left(x,y\right)\hfill & & & \left(x,y\right)\hfill & & & \left(x,y\right)\hfill & & & \left(x,y\right)\hfill \\ \left(+,+\right)\hfill & & & \left(\text{−},+\right)\hfill & & & \left(\text{−},\text{−}\right)\hfill & & & \left(+,\text{−}\right)\hfill \end{array}$
• Points on the Axes
• On the x -axis, $y=0$ . Points with a y -coordinate equal to 0 are on the x -axis, and have coordinates $\left(a,0\right)$ .
• On the y -axis, $x=0$ . Points with an x -coordinate equal to 0 are on the y -axis, and have coordinates $\left(0,b\right).$
• Solution of a Linear Equation
• An ordered pair $\left(x,y\right)$ is a solution of the linear equation $Ax+By=C$ , if the equation is a true statement when the x - and y - values of the ordered pair are substituted into the equation.

## Practice makes perfect

Plot Points in a Rectangular Coordinate System

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

$\left(-4,2\right)$
$\left(-1,-2\right)$
$\left(3,-5\right)$
$\left(-3,5\right)$
$\left(\frac{5}{3},2\right)$ $\left(-2,-3\right)$
$\left(3,-3\right)$
$\left(-4,1\right)$
$\left(4,-1\right)$
$\left(\frac{3}{2},1\right)$

$\left(3,-1\right)$
$\left(-3,1\right)$
$\left(-2,2\right)$
$\left(-4,-3\right)$
$\left(1,\frac{14}{5}\right)$ $\left(-1,1\right)$
$\left(-2,-1\right)$
$\left(2,1\right)$
$\left(1,-4\right)$
$\left(3,\frac{7}{2}\right)$

In the following exercises, plot each point in a rectangular coordinate system.

$\left(-2,0\right)$
$\left(-3,0\right)$
$\left(0,0\right)$
$\left(0,4\right)$
$\left(0,2\right)$ $\left(0,1\right)$
$\left(0,-4\right)$
$\left(-1,0\right)$
$\left(0,0\right)$
$\left(5,0\right)$

$\left(0,0\right)$
$\left(0,-3\right)$
$\left(-4,0\right)$
$\left(1,0\right)$
$\left(0,-2\right)$ $\left(-3,0\right)$
$\left(0,5\right)$
$\left(0,-2\right)$
$\left(2,0\right)$
$\left(0,0\right)$

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system. A: $\left(-4,1\right)$  B: $\left(-3,-4\right)$  C: $\left(1,-3\right)$  D: $\left(4,3\right)$  A: $\left(0,-2\right)$  B: $\left(-2,0\right)$  C: $\left(0,5\right)$  D: $\left(5,0\right)$ Verify Solutions to an Equation in Two Variables

In the following exercises, which ordered pairs are solutions to the given equations?

$2x+y=6$

$\left(1,4\right)$
$\left(3,0\right)$
$\left(2,3\right)$

a, b

$x+3y=9$

$\left(0,3\right)$
$\left(6,1\right)$
$\left(-3,-3\right)$

$4x-2y=8$

$\left(3,2\right)$
$\left(1,4\right)$
$\left(0,-4\right)$

a, c

$3x-2y=12$

$\left(4,0\right)$
$\left(2,-3\right)$
$\left(1,6\right)$

$y=4x+3$

$\left(4,3\right)$
$\left(-1,-1\right)$
$\left(\frac{1}{2},5\right)$

b, c

$y=2x-5$

$\left(0,-5\right)$
$\left(2,1\right)$
$\left(\frac{1}{2},-4\right)$

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

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

a, b

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

$\left(-3,0\right)$
$\left(9,4\right)$
$\left(-6,-1\right)$

Complete a Table of Solutions to a Linear Equation

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

$y=2x-4$

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

$y=3x-1$

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

$y=\text{−}x+5$

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

$y=\text{−}x+2$

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

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

 $x$ $y$ $\left(x,y\right)$ 0 3 6
 $x$ $y$ $\left(x,y\right)$ 0 1 $\left(0,1\right)$ 3 2 $\left(3,2\right)$ 6 3 $\left(6,3\right)$

$y=\frac{1}{2}x+4$

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

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

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

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

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

$x+3y=6$

 $x$ $y$ $\left(x,y\right)$ 0 3 0
 $x$ $y$ $\left(x,y\right)$ 0 2 $\left(0,2\right)$ 3 4 $\left(3,1\right)$ 6 0 $\left(6,0\right)$

$x+2y=8$

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

$2x-5y=10$

 $x$ $y$ $\left(x,y\right)$ 0 10 0
 $x$ $y$ $\left(x,y\right)$ 0 $-2$ $\left(0,-2\right)$ 10 2 $\left(10,2\right)$ 5 0 $\left(5,0\right)$

$3x-4y=12$

 $x$ $y$ $\left(x,y\right)$ 0 8 0

Find Solutions to a Linear Equation

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

$y=5x-8$

$y=3x-9$

$y=-4x+5$

$y=-2x+7$

$x+y=8$

$x+y=6$

$x+y=-2$

$x+y=-1$

$3x+y=5$

$2x+y=3$

$4x-y=8$

$5x-y=10$

$2x+4y=8$

$3x+2y=6$

$5x-2y=10$

$4x-3y=12$

## 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 below, and shown as an ordered pair in the third column.

Plot the points on a coordinate plane. Why is only Quadrant I needed?

 Age $x$ Weight $y$ $\left(x,y\right)$ 0 7 (0, 7) 2 11 (2, 11) 4 15 (4, 15) 6 16 (6, 16) 8 19 (8, 19) 10 20 (10, 20) 12 21 (12, 21) 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 below, and shown as an ordered pair in the third column.

Plot the points on a coordinate plane. Why is only Quadrant I needed?

 Height $x$ Weight $y$ $\left(x,y\right)$ 28 22 (28, 22) 31 27 (31, 27) 33 33 (33, 33) 37 35 (37, 35) 40 41 (40, 41) 42 45 (42, 45)

## Writing exercises

Explain in words how you plot the point $\left(4,-2\right)$ in a rectangular coordinate system.

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

Is the point $\left(-3,0\right)$ on the x -axis or y -axis? How do you know?

Is the point $\left(0,8\right)$ on the x -axis or y -axis? How do you know?

## 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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