Finding Ordered Pair Solutions for Equations with Two Variables

When we solve an equation with two variables, the solution or solutions are ordered pairs. If the variables are x and y, the ordered pair will have the x value first followed by the associated y value. In general, for this section we will be finding some solutions to several equations with two variables by:

1. Choosing the value of one of the variables,
2. Substituting this value into the equation for the associated variable, 
3. Solving the resulting equation for the other variable,
4. Writing this solution as an ordered pair.

Examples:

For each equation, find two ordered pairs that are solutions.

• $y=4x-1$
• $x=\sqrt[]{y-1}$
• $y=x^2-4$

Try to solve this problem on your own, then look below for some possible solutions. Remember, since you are choosing a value for the first variable, there are many more right answers that are not in the solution below.

Solutions:

• $y=4x-1$

If I choose $x=1\:$and $x=0,\:$then when I evaluate $y=4x-1,\:$I will get:

$y=4\left(1\right)-1=4-1=3\:$and $y=4\left(0\right)-1=0-1=-1.$So two ordered pairs that are solutions to $y=4x-1$ are $\left(1,3\right)$ and $\left(0,-1\right).$

• $x=\sqrt[]{y-1}$

If I choose $y=10\:$and $y=17\:$then when I evaluate $x=\sqrt[]{y-1}$,  I will get:

$x=\sqrt[]{10-1}=\sqrt[]{9}=3\:$and $x=\sqrt[]{17-1}=\sqrt[]{16}=4\:$. So two ordered pairs that are solutions to $x=\sqrt[]{y-1}$ are $\left(3,10\right)\:$and $\left(4,17\right).$Always remember to list the x-coordinate first and the y-coordinate second.

• $y=x^2-4$

If I choose $x=0\:$and $x=-1\:$then when I evaluate $y=x^2-4$, I will get:

$y=\left(0\right)^2-4=0-4=-4\:$and $y=\left(-1\right)^2-4=1-4=-3\:$. So two ordered pairs that are solutions to $y=x^2-4$ are $\left(0,-4\right)\:$and $\left(-1,-3\right).$