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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

Tickets for a show are $70 for adults and $50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled $17,200. How many adult tickets and how many child tickets were sold?
Mum Reply
A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
Edi Reply
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?
Jesus Reply
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Ronald Reply
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
Dojzae Reply
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
Xona Reply
5%
Michael
hey everyone how to do algebra
The Reply
Felecia answer 1.5 hours before he reaches her
Adriana Reply
I would like to solve the problem -6/2x
rachel Reply
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year?
Stephanie Reply
Tickets for the community fair cost $12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
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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