3C Creating a Table of Ordered Pair Solutions to Linear Equations

Learning Outcomes

  • Complete a table of values that satisfy a linear equation
  • Find any solution to a linear equation

In the previous examples, 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 we find the ordered pairs if they are not given? One way is to choose a value for x and then solve the equation for y. Or, choose a value for y and then solve for x.

We’ll start by looking at the solutions to the equation y=5x1 we found in the previous chapter. We can summarize this information in a table of solutions.

y=5x1
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.

y=5x1
Substitute x=2 y=5(2)1
Multiply. y=101
Simplify. y=9

The ordered pair is a solution to y=5x1. We will add it to the table.

y=5x1
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 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 an infinite number of solutions for this equation.

example

Complete the table to find three solutions to the equation y=4x2:

y=4x2
x y (x,y)
0
1
2

Solution
Substitute x=0,x=1, and x=2 into y=4x2.

x=0 x=1 x=2
y=4x2 y=4x2 y=4x2
y=402 y=4(1)2 y=422
y=02 y=42 y=82
y=2 y=6 y=6
(0,2) (1,6) (2,6)

The results are summarized in the table.

y=4x2
x y (x,y)
0 2 (0,2)
1 6 (1,6)
2 6 (2,6)

 

try it

 

example

Complete the table to find three solutions to the equation 5x4y=20:

5x4y=20
x y (x,y)
0
0
5

 

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 or y. We could choose 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 as one of our values.

example

Find a solution to the equation 3x+2y=6

 

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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 3x+2y=6.

example

Find three more solutions to the equation 3x+2y=6

 

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Let’s find some solutions to another equation now.

example

Find three solutions to the equation x4y=8.

Remember, there are an infinite number of solutions to each linear equation. Any point you find is a solution if it makes the equation true.

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