Determine the Ordered Pair: Mini-Lesson Review

Mini Lesson Question

Question #1: Determine the Ordered Pair

Which ordered pair is a solution for

y=\frac23x - 4

$(-2, -3)$
$(0, 6)$
$(-3, -2)$
$(6, 0)$

Evaluating Ordered Pairs as Solutions to a Linear Equation

An ordered pair $(x, y)$ is a solution of the linear equation $A x + B y = C$ , 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 $A x + B y = C$ 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 $2 x - y = 3$ .

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

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

This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 9), (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), (5, 7), and (6, 9). The line is labeled y plus 2 x minus 3.

For each ordered pair, decide:

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

( 0, −3)
(3, 3)
(2, −3)
(−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.

Example A shows the ordered pair (0, negative 3). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 3 equals 2 times 0 minus 3. The negative 3 and 0 are colored the same as the negative 3 and 0 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation negative 3 plus negative 3. Below this is the statement (0, negative 3) is a solution. Example B shows the ordered pair (3, 3). Under this is the equation y plus 2 x minus 3. Under this is the equation 3 equals 2 times 3 minus 3. The 3 and 3 are colored the same as the 3 and 3 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation 3 plus 3. Below this is the statement (3, 3) is a solution. Example C shows the ordered pair (2, negative 3). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 3 equals 2 times 2 minus 3. The negative 3 and 2 are colored the same as the negative 3 and 2 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the inequality negative 3 is not equal to 1. Below this is the statement (2, negative 3) is not a solution. Example D shows the ordered pair (negative 1, negative 5). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 5 equals 2 times negative 1 minus 3. The negative 1 and negative 5 are colored the same as the negative 1 and negative 5 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation negative 5 plus negative 5. Below this is the statement (negative 1, negative 5) is a solution.

To answer the second question, examine the graph:

This figure shows the graph of the linear equation y plus 2 x minus 3 and some points graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 1, negative 5), (0, negative 3), and (3, 3). The point (2, negative 3) is also plotted but not on the line.

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

The points that are solutions to $y = 2 x - 3$ 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 $y = 3 x - 1$ . For each ordered pair, decide:

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

(0, −1)
(2, 5)

The figure shows a straight line graphed on the coordinate plane. The x- and y-axes run from negative 10 to 10. The line has arrows on both ends and is labeled y equals three times x minus 1.

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: $(0, - 1)$

$y = 3 x - 1$

$- 1 = 3 (0) - 1$

$- 1 = - 1$

$(0, - 1)$ is a solution

B: $(2, 5)$

$y = 3 x - 1$

$5 = 3 (2) - 1$

$5 = 5$

$(2, 5)$ is a solution

Verifying if specific ordered pairs are solutions to the linear equation y = 3x - 1.

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

Check Your Understanding

Which ordered pair is a solution for $y = - 2 x + 5$ ?

$(1, 2)$
$(- 2, 1)$
$(2, 1)$
$(- 1, 3)$

$(2, 1)$ .

Check yourself: Substituting $2$ for $x$ and $1$ for $y : 1 = - 2 (2) + 5$ ; $1 = - 4 + 5$ ; $1 = 1$ is a true statement.

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.

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