# 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. 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.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
how do you translate this in Algebraic Expressions
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?