Question

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

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

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

Short 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 $\left\{\begin{array}{l}x-5y>10\\ 2x+3y>-2\end{array}\right.$

Step-by-step solution

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.