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

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

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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 3 x + 2 y = 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) ( 1 , 3 2 )
Check.
3 x + 2 y = 6 3 x + 2 y = 6 3 x + 2 y = 6
3 0 + 2 3 6 3 2 + 2 0 6 3 1 + 2 3 2 6
0 + 6 6 6 + 0 6 3 + 3 6
6 = 6 6 = 6 6 = 6

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

3 x + 2 y = 6
x y ( x , y )
0 3 ( 0 , 3 )
2 0 ( 2 , 0 )
1 3 2 ( 1 , 3 2 )

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

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

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Key concepts

  • Sign Patterns of the Quadrants
    Quadrant I Quadrant II Quadrant III Quadrant IV ( x , y ) ( x , y ) ( x , y ) ( x , y ) ( + , + ) ( , + ) ( , ) ( + , )
  • 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 ( a , 0 ) .
    • On the y -axis, x = 0 . Points with an x -coordinate equal to 0 are on the y -axis, and have coordinates ( 0 , b ) .
  • Solution of a Linear Equation
    • An ordered pair ( x , y ) is a solution of the linear equation A x + B y = 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.


( −4 , 2 )
( −1 , −2 )
( 3 , −5 )
( −3 , 5 )
( 5 3 , 2 )

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 2) is plotted and labeled

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( −2 , −3 )
( 3 , −3 )
( −4 , 1 )
( 4 , −1 )
( 3 2 , 1 )

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( 3 , −1 )
( −3 , 1 )
( −2 , 2 )
( −4 , −3 )
( 1 , 14 5 )

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (3, negative 1) is plotted and labeled

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( −1 , 1 )
( −2 , −1 )
( 2 , 1 )
( 1 , −4 )
( 3 , 7 2 )

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


( −2 , 0 )
( −3 , 0 )
( 0 , 0 )
( 0 , 4 )
( 0 , 2 )

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 2, 0) is plotted and labeled

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( 0 , 1 )
( 0 , −4 )
( −1 , 0 )
( 0 , 0 )
( 5 , 0 )

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( 0 , 0 )
( 0 , −3 )
( −4 , 0 )
( 1 , 0 )
( 0 , −2 )

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, 0) is plotted and labeled

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( −3 , 0 )
( 0 , 5 )
( 0 , −2 )
( 2 , 0 )
( 0 , 0 )

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In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 1) is plotted and labeled “A”. The point (negative 3, negative 4) is plotted and labeled “B”. The point (1, negative 3) is plotted and labeled “C”. The point (4, 3) is plotted and labeled “D”.

A: ( −4 , 1 )  B: ( −3 , −4 )  C: ( 1 , −3 )  D: ( 4 , 3 )

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The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, negative 2) is plotted and labeled “A”. The point (negative 2, 0) is plotted and labeled “B”. The point (0, 5) is plotted and labeled “C”. The point (5, 0) is plotted and labeled “D”.

A: ( 0 , −2 )  B: ( −2 , 0 )  C: ( 0 , 5 )  D: ( 5 , 0 )

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Verify Solutions to an Equation in Two Variables

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

2 x + y = 6

( 1 , 4 )
( 3 , 0 )
( 2 , 3 )

a, b

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x + 3 y = 9

( 0 , 3 )
( 6 , 1 )
( −3 , −3 )

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4 x 2 y = 8

( 3 , 2 )
( 1 , 4 )
( 0 , −4 )

a, c

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3 x 2 y = 12

( 4 , 0 )
( 2 , −3 )
( 1 , 6 )

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y = 4 x + 3

( 4 , 3 )
( −1 , −1 )
( 1 2 , 5 )

b, c

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y = 2 x 5

( 0 , −5 )
( 2 , 1 )
( 1 2 , −4 )

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y = 1 2 x 1

( 2 , 0 )
( −6 , −4 )
( −4 , −1 )

a, b

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y = 1 3 x + 1

( −3 , 0 )
( 9 , 4 )
( −6 , −1 )

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Complete a Table of Solutions to a Linear Equation

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

y = 2 x 4

x y ( x , y )
0
2
−1
x y ( x , y )
0 −4 ( 0 , −4 )
2 0 ( 2 , 0 )
−1 −6 ( −1 , −6 )
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y = 3 x 1

x y ( x , y )
0
2
−1
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y = x + 5

x y ( x , y )
0
3
−2
x y ( x , y )
0 5 ( 0 , 5 )
3 2 ( 3 , 2 )
−2 7 ( −2 , 7 )
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y = x + 2

x y ( x , y )
0
3
−2
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y = 1 3 x + 1

x y ( x , y )
0
3
6
x y ( x , y )
0 1 ( 0 , 1 )
3 2 ( 3 , 2 )
6 3 ( 6 , 3 )
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y = 1 2 x + 4

x y ( x , y )
0
2
4
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y = 3 2 x 2

x y ( x , y )
0
2
−2
x y ( x , y )
0 −2 ( 0 , −2 )
2 −5 ( 2 , −5 )
−2 1 ( −2 , 1 )
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y = 2 3 x 1

x y ( x , y )
0
3
−3
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x + 3 y = 6

x y ( x , y )
0
3
0
x y ( x , y )
0 2 ( 0 , 2 )
3 4 ( 3 , 1 )
6 0 ( 6 , 0 )
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x + 2 y = 8

x y ( x , y )
0
4
0
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2 x 5 y = 10

x y ( x , y )
0
10
0
x y ( x , y )
0 −2 ( 0 , −2 )
10 2 ( 10 , 2 )
5 0 ( 5 , 0 )
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3 x 4 y = 12

x y ( x , y )
0
8
0
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Find Solutions to a Linear Equation

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

y = 5 x 8

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y = −4 x + 5

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x + y = 8

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x + y = −2

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3 x + y = 5

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4 x y = 8

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2 x + 4 y = 8

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5 x 2 y = 10

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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 ( x , y )
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)


The graph shows the x y-coordinate plane. The x- and y-axes each run from 0 to 25. The points (0, 7), (2, 11), (4, 15), (6, 16), (8, 19), (10, 20) and (12, 21) are plotted and labeled.
Age and weight are only positive.

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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 ( x , y )
28 22 (28, 22)
31 27 (31, 27)
33 33 (33, 33)
37 35 (37, 35)
40 41 (40, 41)
42 45 (42, 45)
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Writing exercises

Explain in words how you plot the point ( 4 , −2 ) in a rectangular coordinate system.

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

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Is the point ( −3 , 0 ) on the x -axis or y -axis? How do you know?

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

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Self check

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

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “plot points in a rectangular coordinate system,”, “identify points on a graph,” “verify solutions to an equation in two variables,” “complete a table of solutions to a linear equation,” and “find solutions to a linear equation.” The rest of the cells are blank.

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.

Questions & Answers

3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
Edwin Reply
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
DaYoungan Reply
p-2/3=5/6 how do I solve it with explanation pls
Adedamola Reply
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
don't understand answer
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
Adedamola Reply
yes
James
don't understand answer
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy
He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
32
Cindy
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
Eric here. I'm a parent. 53 years old. I have never taken algebra. I want to learn.
Eric
I am 63 and never learned algebra
Cindy
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
they are 92 candies in the bag
POP
92
Cindy
rectangular field solutions
Navin Reply
What is this?
Donna
t
muqtaar
Practice Key Terms 7

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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