An ordered pair [latex]\left(x,y\right)[/latex] is a solution of the linear equation [latex]ax+by=c[/latex], if the equation is a true statement when the [latex]x[/latex]– and [latex]y[/latex]-values of the ordered pair are substituted into the equation.
Example
Determine whether [latex](−2,4)[/latex] is a solution of the equation [latex]4y+5x=3[/latex].
Solution
Substitute [latex]x=−2[/latex] and [latex]y=4[/latex] into the equation:
The statement is not true, so [latex](−2,4)[/latex] is not a solution.
Answer
[latex](−2,4)[/latex] is not a solution of the equation [latex]4y+5x=3[/latex].
example
Determine which ordered pairs are solutions of the equation [latex]x+4y=8\text<:>[/latex]
1. [latex]\left(0,2\right)[/latex]
2. [latex]\left(2,-4\right)[/latex]
3. [latex]\left(-4,3\right)[/latex]
Solution
Substitute the [latex]x\text<- and >y\text[/latex] from each ordered pair into the equation and determine if the result is a true statement.->
[latex]\left(2,-4\right)[/latex] is not a solution.
[latex]\left(-4,3\right)[/latex] is a solution.
try it
example
Determine which ordered pairs are solutions of the equation. [latex]y=5x - 1\text<:>[/latex]
1. [latex]\left(0,-1\right)[/latex]
2. [latex]\left(1,4\right)[/latex]
3. [latex]\left(-2,-7\right)[/latex]
Solution
Substitute the [latex]x\text<->[/latex] and [latex]y\text[/latex] from each ordered pair into the equation and determine if it results in a true statement.->
In the previous examples, we substituted the [latex]x\text<- and >y\text[/latex] of a given ordered pair to determine whether or not it was a solution of a given linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for [latex]x[/latex] and then solve the equation for [latex]y[/latex]. Or, choose a value for [latex]y[/latex] and then solve for [latex]x[/latex].->
Let’s consider the equation [latex]y=5x - 1[/latex]. The easiest value to choose for [latex]x[/latex] or [latex]y[/latex] is zero:
We can continue to find more solutions by choosing different values of [latex]x[/latex] and [latex]y[/latex].