Q 4.106 - Intermediate Algebra

Question

Determine whether the ordered pair is the solution to the system: $\left\{\begin{array}{l}y>4x-2\\ 4x-y<20\end{array}\right.$

$a\left)\right(-2,1)$

$b\left)\right(4,-1)$

Step-by-Step Solution

Answer

The ordered pair $(-2,1)$ is a solution to the system of inequalities $\left\{\begin{array}{l}y>4x-2\\ 4x-y<20\end{array}\right.$

And the ordered pair $(4,-1)$ is not a solution to the system of inequalities $\left\{\begin{array}{l}y>4x-2\\ 4x-y<20\end{array}\right.$

Part a ) Step 1. Given

The system of inequalities, $\left\{\begin{array}{l}y>4x-2\\ 4x-y<20\end{array}\right.$

$(x,y)=(-2,1)$

$(x,y)=(4,-1)$

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

Substitute the point $(-2,1)$ in the system of inequalities.

   $y>4x-2$

$\left(1\right)>4(-2)-2$

   $1>-10$ holds true.

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

Substitute the point $(-2,1)$ in the system of inequalities.

         $4x-y<20$

$4(-2)-\left(1\right)<20$

             $-9<20$ holds true.

The ordered pair $(-2,1)$ made both the inequality true.

Therefore, $(-2,1)$ is a solution to this system.

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

Substitute the point $(4,-1)$ in the system of inequality

      $y>4x-2$

$(-1)>4\left(4\right)-2$

    $-1>14$ is false

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

Substitute the point $(4,-1)$ in the system of inequalities.

         $4x-y<20$

$4\left(4\right)-(-1)<20$

        $16+1<20$

              $17<20$ is true

The ordered pair $(4,-1)$ made one inequality true, but the other one is false.

Therefore, $(4,-1)$ is not a solution to this system.