Completing 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 \(y=5x-1\) that we found in the last example from the previous lesson. We can summarize this information in a table of solutions, as shown in the table below.

\(y=5x-1\)
\(x\)	\(y\)	\(\left(x,y\right)\)
0	\(-1\)	\(\left(0,-1\right)\)
1	4	\(\left(1,4\right)\)

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 \(\left(2,9\right)\) is a solution to \(y=5x-1\). We will add it to the table below.

\(y=5x-1\)
\(x\)	\(y\)	\(\left(x,y\right)\)
0	\(-1\)	\(\left(0,-1\right)\)
1	4	\(\left(1,4\right)\)
2	9	\(\left(2,9\right)\)

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.

Example

Complete the table below to find three solutions to the equation \(y=4x-2\).

\(y=4x-2\)
\(x\)	\(y\)	\(\left(x,y\right)\)
0
\(-1\)
2

Solution

Substitute \(x=0\), \(x=-1\), and \(x=2\) into \(y=4x-2\).

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 the table below.

\(y=4x-2\)
\(x\)	\(y\)	\(\left(x,y\right)\)
0	\(-2\)	\(\left(0,-2\right)\)
\(-1\)	\(-6\)	\(\left(-1,-6\right)\)
2	6	\(\left(2,6\right)\)

Example

Complete the table below to find three solutions to the equation \(5x-4y=20\).

\(5x-4y=20\)
\(x\)	\(y\)	\(\left(x,y\right)\)
0
	0
	5

Solution

Substitute the given value into the equation \(5x-4y=20\) 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 the table below.

\(5x-4y=20\)
\(x\)	\(y\)	\(\left(x,y\right)\)
0	\(-5\)	\(\left(0,-5\right)\)
4	0	\(\left(4,0\right)\)
8	5	\(\left(8,5\right)\)

Completing a Table of Solutions to a Linear Equation in Two Variables