Mathematics » Introducing Graphs » Use the Rectangular Coordinate System

# Completing a Table of Solutions to a Linear Equation

## Completing a Table of Solutions to a Linear Equation

In the previous examples, we substituted the $$x\text{- and}\phantom{\rule{0.2em}{0ex}}y\text{-values}$$ 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 $$y=5x-1$$ we found in the last example in the previous lesson. We can summarize this information in a table of solutions.

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

To find a third solution, we’ll let $$x=2$$ and solve for $$y.$$

 $$y=5x-1$$ Multiply. $$y=10-1$$ Simplify. $$y=9$$

The ordered pair is a solution to $$y=5x-1$$. We will add it to the table.

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

We can find more solutions to the equation by substituting 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 $$y=4x-2\text{:}$$

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

### Solution

Substitute $$x=0,x=-1,$$ and $$x=2$$ into $$y=4x-2.$$

 $$y=4x-2$$ $$y=4x-2$$ $$y=4x-2$$ $$y=0-2$$ $$y=-4-2$$ $$y=8-2$$ $$y=-2$$ $$y=-6$$ $$y=6$$ $$\left(0,-2\right)$$ $$\left(-1,-6\right)$$ $$\left(2,6\right)$$

The results are summarized in the table.

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

## Example

Complete the table to find three solutions to the equation $$5x-4y=20\text{:}$$

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

### Solution

The results are summarized in the table.

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

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