Verifying Solutions to an Equation in Two Variables
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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