Determine the Ordered Pair: Mini-Lesson Review

Evaluating Ordered Pairs as Solutions to a Linear Equation 

An ordered pair $(x,y)$ is a solution of the linear equation $Ax+By=C$, if the equation is a true statement when the x- and $y$-values of the ordered pair are substituted into the equation.

Linear equations have infinitely many solutions. We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We connect the points with a straight line to get the graph of the equation. We put arrows on the ends of each side of the line to indicate that the line continues in both directions.

A graph is a visual representation of all the solutions of the equation. It is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to that equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation. Points not on the line are not solutions!

The graph of a linear equation $Ax+By=C$ is a straight line.

• Every point on the line is a solution of the equation.
• Every solution of this equation is a point on this line.

Let's examine the equation $2x-y=3$.

To analyze the equation, we may want to graph the line. If we change the equation to slope-intercept form, the equation is $y=2x-3$.

The graph of $y=2x-3$ is shown.

For each ordered pair, decide:

• Is the ordered pair a solution to the equation?
• Is the point on the line?

1. ( 0, −3)
2. (3, 3)
3. (2, −3)
4. (−1, −5)

To answer the first question, substitute the $x-$ and $y-$ values into the equation to check if the ordered pair is a solution to the equation.

To answer the second question, examine the graph:

The points $(0,3)$, $(3,-3)$, and $(-1,-5)$ are on the line $y=2x-3$, and the point $(2,-3)$ is not on the line.

The points that are solutions to $y=2x-3$ are on the line, but the point that is not a solution is not on the line.

Try it

Evaluating Ordered Pairs as Solutions to a Linear Equation 

Use the graph of $y=3x-1$. For each ordered pair, decide:

• Is the ordered pair a solution to the equation?
• Is the point on the line?

1. (0, −1)
2. (2, 5)

Here is how to answer the first question:

Substitute the x- and$y$-values into the equation to check if the ordered pair is a solution to the equation.

A: $(0,-1)$ $y=3x-1$ $-1=3(0)-1$ $-1=-1$ $(0,-1)$ is a solution

B: $(2,5)$ $y=3x-1$ $5=3(2)-1$ $5=5$ $(2,5)$ is a solution

To answer the second question, the points are on the line since they are solutions to $y=3x-1$.

Video: Ordered Pairs and Solutions to 2-Variable Equations

Khan Academy: Solutions to 2-Variable Equations

Watch this video to learn about ordered pairs and how to test solutions to 2-variable equations.