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Which of the following ordered pairs are solutions to the equation y = 4 x 3 ?

( 0 , 3 ) ( 1 , 1 ) ( −1 , −1 )

b

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Which of the following ordered pairs are solutions to the equation y = −2 x + 6 ?

( 0 , 6 ) ( 1 , 4 ) ( −2 , −2 )

a, b

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Complete a table of solutions to a linear equation in two variables

In the examples above, we substituted the x - and y -values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do you find the ordered pairs if they are not given? It’s easier than you might think—you can just pick a value for x and then solve the equation for y . Or, pick a value for y and then solve for x .

We’ll start by looking at the solutions to the equation y = 5 x 1 that we found in [link] . We can summarize this information in a table of solutions, as shown in [link] .

y = 5 x 1
x y ( x , y )
0 −1 ( 0 , −1 )
1 4 ( 1 , 4 )

To find a third solution, we’ll let x = 2 and solve for y .

The figure shows the steps to solve for y when x equals 2 in the equation y equals 5 x minus 1. The equation y equals 5 x minus 1 is shown. Below it is the equation with 2 substituted in for x which is y equals 5 times 2 minus 1. To solve for y first multiply so that the equation becomes y equals 10 minus 1 then subtract so that the equation is y equals 9.

The ordered pair ( 2 , 9 ) is a solution to y = 5 x 1 . We will add it to [link] .

y = 5 x 1
x y ( x , y )
0 −1 ( 0 , −1 )
1 4 ( 1 , 4 )
2 9 ( 2 , 9 )

We can find more solutions to the equation by substituting in any value of x or any value of y and solving the resulting equation to get another ordered pair that is a solution. There are infinitely many solutions of this equation.

Complete [link] to find three solutions to the equation y = 4 x 2 .

y = 4 x 2
x y ( x , y )
0
−1
2

Solution

Substitute x = 0 , x = −1 , and x = 2 into y = 4 x 2 .
This figure has three columns. At the top of the first column is the value x equals 0. Below this is the equation y equals 4x minus 2. Below this is the same equation with 0 substituted for x: y equals 4 times 0 minus 2. Below this is y equals 0 minus 2. Below this is y equals negative 2. Below this is the ordered pair (0, negative 2). At the top of the second column is the value x equals negative 1. Below this is the equation y equals 4x minus 2. Below this is the same equation with negative 1 substituted for x: y equals 4 times minus 1 minus 2. Below this is y equals negative 4 minus 2. Below this is y equals negative 6. Below this is the ordered pair (negative 1, negative 6). At the top of the third column is the value x equals 2. Below this is the equation y equals 4x minus 2. Below this is the same equation with 2 substituted for x: y equals 4 times 2 minus 2. Below this is y equals 8 minus 2. Below this is y equals 6. Below this is the ordered pair (2, 6).

The results are summarized in [link] .

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

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Complete the table to find three solutions to this equation: y = 3 x 1 .

y = 3 x 1
x y ( x , y )
0
−1
2
y = 3 x 1
x y ( x , y )
0 −1 ( 0 , −1 )
−1 −4 ( −1 , −4 )
2 5 ( 2 , 5 )
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Complete the table to find three solutions to this equation: y = 6 x + 1 .

y = 6 x + 1
x y ( x , y )
0
1
−2
y = 6 x + 1
x y ( x , y )
0 1 ( 0 , 1 )
1 7 ( 1 , 7 )
−2 −11 ( −2 , −11 )
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Complete [link] to find three solutions to the equation 5 x 4 y = 20 .

5 x 4 y = 20
x y ( x , y )
0
0
5

Solution

Substitute the given value into the equation 5 x 4 y = 20 and solve for the other variable. Then, fill in the values in the table.
This figure has three columns. At the top of the first column is the value x equals 0. Below this is the equation 5x minus 4y equals 20. Below this is the same equation with 0 substituted for x: 5 times 0 minus 4y equals 20. Below this is 0 minus 4y equals 20. Below this is negative 4y equals 20. Below this is y equals negative 5. Below this is the ordered pair (0, negative 5). At the top of the second column is the value y equals 0. Below this is the equation 5x minus 4y equals 20. Below this is the same equation with 0 substituted for y: 5x minus 4 times 0 equals 20. Below this is 5x minus 0 equals 20. Below this is 5x equals 20. Below this is x equals 4. Below this is the ordered pair (4, 0). At the top of the third column is the value y equals 5. Below this is the equation 5x minus 47 equals 20. Below this is the same equation with 5 substituted for y: 5x minus 4 times 5 equals 20. Below this is the equation 5x minus 20 equals 20. Below this is 5x equals 40. Below this is x equals 8. Below this is the ordered pair (8, 5).

The results are summarized in [link] .

5 x 4 y = 20
x y ( x , y )
0 −5 ( 0 , −5 )
4 0 ( 4 , 0 )
8 5 ( 8 , 5 )

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Complete the table to find three solutions to this equation: 2 x 5 y = 20 .

2 x 5 y = 20
x y ( x , y )
0
0
−5
2 x 5 y = 20
x y ( x , y )
0 −4 ( 0 , −4 )
10 0 ( 10 , 0 )
−5 −6 ( −5 , −6 )
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Complete the table to find three solutions to this equation: 3 x 4 y = 12 .

3 x 4 y = 12
x y ( x , y )
0
0
−4
3 x 4 y = 12
x y ( x , y )
0 −3 ( 0 , −3 )
4 0 ( 4 , 0 )
−4 −6 ( −4 , −6 )
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Find solutions to a linear equation

To find a solution to a linear equation, you really can pick any number you want to substitute into the equation for x or y . But since you’ll need to use that number to solve for the other variable it’s a good idea to choose a number that’s easy to work with.

When the equation is in y -form, with the y by itself on one side of the equation, it is usually easier to choose values of x and then solve for y .

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

Solution

We can substitute any value we want for x or any value for y . Since the equation is in y -form, it will be easier to substitute in values of x . Let’s pick x = 0 , x = 1 , and x = −1 .

. . .
. . .
Substitute the value into the equation. . . .
Simplify. . . .
Simplify. . . .
Write the ordered pair. (0, 2) (1, −1) (−1, 5)
Check.
y = −3 x + 2 y = −3 x + 2 y = −3 x + 2
2 −3 0 + 2 −1 −3 1 + 2 5 −3 ( −1 ) + 2
2 0 + 2 −1 −3 + 2 5 3 + 2
2 = 2 −1 = −1 5 = 5

So, ( 0 , 2 ) , ( 1 , −1 ) and ( −1 , 5 ) are all solutions to y = −3 x + 2 . We show them in [link] .

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

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Questions & Answers

The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Edi Reply
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?
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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?
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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
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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
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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
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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
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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
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Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
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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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