<< Chapter < Page Chapter >> Page >

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

( 0 , 3 ) ( 1 , 1 ) ( −1 , −1 )

b

Got questions? Get instant answers now!

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

( 0 , 6 ) ( 1 , 4 ) ( −2 , −2 )

a, b

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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 )
Got questions? Get instant answers now!

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 )
Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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 )
Got questions? Get instant answers now!

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 )
Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!
Got questions? Get instant answers now!
Practice Key Terms 7

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask