## 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.

\(5x-4y=20\) | ||
---|---|---|

\(x\) | \(y\) | \(\left(x,y\right)\) |

\(0\) | \(-5\) | \(\left(0,-5\right)\) |

\(4\) | \(0\) | \(\left(4,0\right)\) |

\(8\) | \(5\) | \(\left(8,5\right)\) |