Q 4.105 - Intermediate Algebra

Question

Determine whether the ordered pair is the solution to the system: $\left\{\begin{array}{l}x-5y>10\\ 2x+3y>-2\end{array}\right.$

$a\left)\right(3,-1)$

$b\left)\right(6,-3)$

Step-by-Step Solution

Answer

The ordered pair $(3,-1)$ is not a solution to the system of inequalities $\left\{\begin{array}{l}x-5y>10\\ 2x+3y>-2\end{array}\right.$

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

Part a ) Step 1. Given information

The system of inequalities, $\left\{\begin{array}{l}x-5y>10\\ 2x+3y>-2\end{array}\right.$

$(x,y)=(3,-1)$

Part a ) Step 2. Substitute the points in the first equation

Substitute the point $(3,-1)$ in the system of inequalities.

         $x-5y>10$

$\left(3\right)-5(-1)>10$

           $3+5>10$

                $8>10$

Hence the equation does not hold true

Part a ) Step 3. Substitute the points in the second equation

Substitute the point $(3,-1)$ in the system of inequalities.

        $2x+3y>-2$

$2\left(3\right)+3(-1)>-2$

             $6-3>-2$

                  $3>-2$ which is true.

The ordered pair $(3,-1)$ made one inequality true, but the other one is false.

Therefore, $(3,-1)$ is not a solution to this system.

Part b ) Step 1. Substitute the points in the first equation

Substitute the point $(6,-3)$ in the system of inequality

         $x-5y>10$

$\left(6\right)-5(-3)>10$

        $6+15>10$

             $21>10$ holds true.

Part b ) Step 2. Substitute the points in the second equation

Substitute the point $(6,-3)$ in the system of inequalities.

        $2x+3y>-2$

$2\left(6\right)+3(-3)>-2$

    $12+(-9)>-2$

                   $3>-2$ holds true.

The ordered pair $(6,-3)$ made both the inequality true.

Therefore, $(6,-3)$ is a solution to this system.