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Which of the following ordered pairs are solutions to the equation ?
ⓐ ⓑ ⓒ
b
Which of the following ordered pairs are solutions to the equation ?
ⓐ ⓑ ⓒ
a, b
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 and then solve the equation for . Or, pick a value for and then solve for .
We’ll start by looking at the solutions to the equation that we found in [link] . We can summarize this information in a table of solutions, as shown in [link] .
0 | ||
1 | 4 |
To find a third solution, we’ll let and solve for .
The ordered pair is a solution to . We will add it to [link] .
0 | ||
1 | 4 | |
2 | 9 |
We can find more solutions to the equation by substituting in any value of or any value of 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 .
0 | ||
2 |
Complete the table to find three solutions to this equation: .
0 | ||
2 |
0 | ||
2 | 5 |
Complete the table to find three solutions to this equation: .
0 | ||
1 | ||
0 | 1 | |
1 | 7 | |
Complete [link] to find three solutions to the equation .
0 | ||
0 | ||
5 |
Substitute the given value into the equation
and solve for the other variable. Then, fill in the values in the table.
The results are summarized in [link] .
0 | ||
4 | 0 | |
8 | 5 |
Complete the table to find three solutions to this equation: .
0 | ||
0 | ||
0 | ||
10 | 0 | |
Complete the table to find three solutions to this equation: .
0 | ||
0 | ||
0 | ||
4 | 0 | |
To find a solution to a linear equation, you really can pick any number you want to substitute into the equation for or 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 and then solve for .
Find three solutions to the equation .
We can substitute any value we want for
or any value for
. Since the equation is in
y -form, it will be easier to substitute in values of
. Let’s pick
,
, and
.
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Substitute the value into the equation. |
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Simplify. |
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Simplify. |
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Write the ordered pair. | (0, 2) | (1, −1) | (−1, 5) | ||
Check. | |||||
So,
,
and
are all solutions to
. We show them in
[link] .
0 | 2 | |
1 | ||
5 |
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