find three ordered pairs (x,y) such that 2x - 5y = 30 and plot them.
Answers
Step-by-step explanation:
In the previous examples, we substituted the
x
- and
y
-values
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 we find the ordered pairs if they are not given? One way is to choose a value for
x
x and then solve the equation for
y
y. Or, choose a value for
y
y and then solve for
x
x.
We’ll start by looking at the solutions to the equation
y
=
5
x
−
1
y=5x−1 we found in the previous chapter. We can summarize this information in a table of solutions.
y
=
5
x
−
1
y=5x−1
x
x
y
y
(
x
,
y
)
(x,y)
0
0
−
1
−1
(
0
,
−
1
)
(0,−1)
1
1
4
4
(
1
,
4
)
(1,4)
To find a third solution, we’ll let
x
=
2
x=2 and solve for
y
y.
y
=
5
x
−
1
y=5x−1
.
y
=
5
(
2
)
−
1
y=5(2)−1
Multiply.
y
=
10
−
1
y=10−1
Simplify.
y
=
9
y=9
The ordered pair is a solution to
y
=
5
x
−
1
y=5x−1. We will add it to the table.
y
=
5
x
−
1
y=5x−1
x
x
y
y
(
x
,
y
)
(x,y)
0
0
−
1
−1
(
0
,
−
1
)
(0,−1)
1
1
4
4
(
1
,
4
)
(1,4)
2
2
9
9
(
2
,
9
)
(2,9)
We can find more solutions to the equation by substituting any value of
x
x or any value of
y
y and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.
EXAMPLE
Complete the table to find three solutions to the equation
y
=
4
x
−
2
:
y=4x−2:
y
=
4
x
−
2
y=4x−2
x
x
y
y
(
x
,
y
)
(x,y)
0
0
−
1
−1
2
2
Solution
Substitute
x
=
0
,
x
=
−
1
x=0,x=−1, and
x
=
2
x=2 into
y
=
4
x
−
2
y=4x−2.
x
=
0
x=0
x
=
−
1
x=−1
x
=
2
x=2
y
=
4
x
−
2
y=4x−2
y
=
4
x
−
2
y=4x−2
y
=
4
x
−
2
y=4x−2
y
=
4
⋅
0
−
2
y=4⋅0−2
y
=
4
(
−
1
)
−
2
y=4(−1)−2
y
=
4
⋅
2
−
2
y=4⋅2−2
y
=
0
−
2
y=0−2
y
=
−
4
−
2
y=−4−2
y
=
8
−
2
y=8−2
y
=
−
2
y=−2
y
=
−
6
y=−6
y
=
6
y=6
(
0
,
−
2
)
(0,−2)
(
−
1
,
−
6
)
(−1,−6)
(
2
,
6
)
(2,6)
The results are summarized in the table.
y
=
4
x
−
2
y=4x−2
x
x
y
y
(
x
,
y
)
(x,y)
0
0
−
2
−2
(
0
,
−
2
)
(0,−2)
−
1
−1
−
6
−6
(
−
1
,
−
6
)
(−1,−6)
2
2
6
6
(
2
,
6
)
(2,6)
TRY IT
EXAMPLE
Complete the table to find three solutions to the equation
5
x
−
4
y
=
20
:
5x−4y=20:
5
x
−
4
y
=
20
5x−4y=20
x
x
y
y
(
x
,
y
)
(x,y)
0
0
0
0
5
5
Show Solution
TRY IT
Find Solutions to Linear Equations in Two Variables
To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either
x
x or
y
y. We could choose
1
,
100
,
1
,
000
1,100,1,000, or any other value we want. But it’s a good idea to choose a number that’s easy to work with. We’ll usually choose
0
0 as one of our values.
EXAMPLE
Find a solution to the equation
3
x
+
2
y
=
6
3x+2y=6.
Show Solution
TRY IT
We said that linear equations in two variables have infinitely many solutions, and we’ve just found one of them. Let’s find some other solutions to the equation
3
x
+
2
y
=
6
3x+2y=6.