Q. 398 - Intermediate Algebra

Question

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system.

$\left\{\begin{array}{l}y>\frac{1}{3}x+2\\ x-\frac{1}{4}y\le 10\end{array}\right.$

(a) $(6,5)$

(b) $(15,8)$

Step-by-Step Solution

Answer

The ordered pair $(6,5)$ is a solution to the system of inequalities $\left\{\begin{array}{l}y>\frac{1}{3}x+2\\ x-\frac{1}{4}y\le 10\end{array}\right.$

The ordered pair $(15,8)$ is not a solution to the system of inequality $\left\{\begin{array}{l}y>\frac{1}{3}x+2\\ x-\frac{1}{4}y\le 10\end{array}\right.$

Part (a) Step 1. Given

The system of inequalities $\left\{\begin{array}{l}y>\frac{1}{3}x+2\\ x-\frac{1}{4}y\le 10\end{array}\right.$

To find if the ordered pair $(6,5)$ is a solution to the given inequality.

Part (a) Step 2. Substitute the points in the first inequality

Substitute $(6,5)$ in the inequality,

$y>\frac{1}{3}x+2$

$5>\frac{1}{3}\left(6\right)+2$

$5>4$ is true.

Part (a) Step 3. Substitute the point in the second inequality

Substitute $(6,5)$ in the inequality $x-\frac{1}{4}y\le 10$,

$6-\frac{1}{4}\left(5\right)\le 10$

  $\frac{24-5}{4}\le 10$

       $\frac{19}{4}\le 10$

         $19\le 40$ is true.

The ordered pair $(6,5)$ made both the inequality true.

So the ordered pair $(6,5)$ is a solution to the inequality.

Part (b) Step 1. Given

The system of inequalities $\left\{\begin{array}{l}y>\frac{1}{3}x+2\\ x-\frac{1}{4}y\le 10\end{array}\right.$

To find if the ordered pair $(15,8)$ is a solution to the given inequality.

Part (b) Step 2. Substitute the point in the first inequality

Substitute $(15,8)$ in the inequality,

$y>\frac{1}{3}x+2$

$8>\frac{1}{3}\left(15\right)+2$

$8>5+2$

$8>7$ is true.

Part (b) Step 3. Substitute the points in the second inequality

Substitute $(15,8)$ in the inequality,

     $x-\frac{1}{4}y\le 10$

$15-\frac{1}{4}\left(8\right)\le 10$

      $15-2\le 10$

             $13\le 10$ is false.

So the ordered pair $(15,8)$ made one of the inequality true and the other one false.

So the ordered pair $(15,8)$ is not a solution to the system of inequalities.