Question

35 solve the following syste
wing system of inequalities graphically on the grid below and label the solution S.
3x + 4y > 20
x < 3y - 18
Is the point (3,7) in the solution set? Explain your answer.

Answer 1

Part A: The solution is $(-0.923,5.692)$

Part B: The point $(3,7)$ is not in the solution set.

Explanation:

Part A: The given inequalities are $3 x+4 y>20$ and $x<3 y-18$

The solution can be determined by solving the two inequalities by substitution method.

Changing inequalities to equality, we have,

$x=3 y-18$ and $3 x+4 y=20$

Let us substitute $x=3 y-18$ in the equation $3 x+4 y=20$ , we get,

$3 (3y-18)+4 y=20$

   $9y-54+4y=20$

                  $13y=74$

                     $y=5.692$

Substituting $y=5.692$ in $x=3 y-18$, we get,

$x=3 (5.692)-18$

  $=17.076-18$

$x=-0.923$

Thus, the solution set is $(-0.923,5.692)$

Part B: Now, we shall determine whether the point $(3,7)$ is in the solution set.

Let us substitute the point $(3,7)$ in the inequalities $3 x+4 y>20$ and $x<3 y-18$, we get,

$3 (3)+4 (7)>20$

      $9+28>20$

            $37>20$

Also, substituting $(3,7)$ in $x<3 y-18$, we get,

$3<3 (7)-18$

$3<21-18$

$3<3$

Since, the point $(3,7)$ does not satisfy one of the inequality $x<3 y-18$ , the solution set does not contain the point $(3,7)$

Thus, the point $(3,7)$ is not in the solution set.