# 11.1 Use the rectangular coordinate system  (Page 6/13)

 Page 6 / 13

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. 2. 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.

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?      By Qqq Qqq By By  By Rhodes