## Verifying Solutions to an Equation in Two Variables

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All the equations we solved so far have been equations with one variable. In almost every case, when we solved the equation we got exactly one **solution**. The process of solving an equation ended with a statement such as \(x=4.\) Then we checked the solution by substituting back into the equation.

Here’s an example of a **linear equation in one variable**, and its one solution.

But equations can have more than one variable. Equations with two variables can be written in the general form \(Ax+By=C.\) An equation of this form is called a linear equation in two variables.

### Definition: Linear Equation

An equation of the form \(Ax+By=C,\) where \(A\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}B\) are not both zero, is called a linear equation in two variables.

Notice that the word “line” is in linear.

Here is an example of a linear equation in two variables, \(x\) and \(y\text{:}\)

Is \(y=-5x+1\) a linear equation? It does not appear to be in the form \(Ax+By=C.\) But we could rewrite it in this form.

Add \(5x\) to both sides. | |

Simplify. | |

Use the Commutative Property to put it in \(Ax+By=C.\) |

By rewriting \(y=-5x+1\) as \(5x+y=1,\) we can see that it is a linear equation in two variables because it can be written in the form \(Ax+By=C.\)

Linear equations in two variables have infinitely many solutions. For every number that is substituted for \(x,\) there is a corresponding \(y\) value. This pair of values is a **solution to the linear equation** and is represented by the ordered pair \(\left(x,y\right).\) When we substitute these values of \(x\) and \(y\) into the equation, the result is a true statement because the value on the left side is equal to the value on the right side.

### Definition: Solution to a Linear Equation in Two Variables

An ordered pair \(\left(x,y\right)\) is a solution to the linear equation \(Ax+By=C,\) if the equation is a true statement when the \(x\text{-}\) and \(y\text{-values}\) of the ordered pair are substituted into the equation.

## Example

Determine which ordered pairs are solutions of the equation \(x+4y=8\text{:}\)

- \(\phantom{\rule{0.2em}{0ex}}\left(0,2\right)\)
- \(\phantom{\rule{0.2em}{0ex}}\left(2,-4\right)\)
- \(\phantom{\rule{0.2em}{0ex}}\left(-4,3\right)\)

### Solution

Substitute the \(x\text{- and}\phantom{\rule{0.2em}{0ex}}y\text{-values}\) from each ordered pair into the equation and determine if the result is a true statement.

\(\phantom{\rule{0.2em}{0ex}}\left(0,2\right)\) | \(\phantom{\rule{0.2em}{0ex}}\left(2,-4\right)\) | \(\phantom{\rule{0.2em}{0ex}}\left(-4,3\right)\) |

\(\left(0,2\right)\) is a solution. | \(\left(2,-4\right)\) is not a solution. | \(\left(-4,3\right)\) is a solution. |

## Example

Determine which ordered pairs are solutions of the equation. \(y=5x-1\text{:}\)

- \(\phantom{\rule{0.2em}{0ex}}\left(0,-1\right)\)
- \(\phantom{\rule{0.2em}{0ex}}\left(1,4\right)\)
- \(\phantom{\rule{0.2em}{0ex}}\left(-2,-7\right)\)

### Solution

Substitute the \(x\text{-}\) and \(y\text{-values}\) from each ordered pair into the equation and determine if it results in a true statement.

\(\phantom{\rule{0.2em}{0ex}}\left(0,-1\right)\) | \(\phantom{\rule{0.2em}{0ex}}\left(1,4\right)\) | \(\phantom{\rule{0.2em}{0ex}}\left(-2,-7\right)\) |

\(\left(0,-1\right)\) is a solution. | \(\left(1,4\right)\) is a solution. | \(\left(-2,-7\right)\) is not a solution. |

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