Questions: Determine whether the given point is in the feasible set of this system of inequalities. 9x + 3y ≤ 82 x + y ≤ 17 2x + 6y ≤ 66 ;(6,7) x ≥ 0 y ≥ 0 Choose the correct answer below. No Yes

Determine whether the given point is in the feasible set of this system of inequalities.

9x + 3y ≤ 82
x + y ≤ 17
2x + 6y ≤ 66 ;(6,7)
x ≥ 0
y ≥ 0

Choose the correct answer below.
No
Yes

Transcript text: Determine whether the given point is in the feasible set of this system of inequalities. \[ \left\{\begin{aligned} 9 x+3 y & \leq 82 \\ x+y & \leq 17 \\ 2 x+6 y & \leq 66 ;(6,7) \\ x & \geq 0 \\ y & \geq 0 \end{aligned}\right. \] Choose the correct answer below. No Yes

Solution

Solution Steps

To determine if the point (6, 7) is in the feasible set of the given system of inequalities, we need to check if it satisfies all the inequalities. We will substitute \( x = 6 \) and \( y = 7 \) into each inequality and verify if all conditions hold true.

Step 1: Check Inequality 1

We evaluate the first inequality: \[ 9x + 3y \leq 82 \] Substituting \( x = 6 \) and \( y = 7 \): \[ 9(6) + 3(7) = 54 + 21 = 75 \leq 82 \] This inequality holds true.

Step 2: Check Inequality 2

Next, we evaluate the second inequality: \[ x + y \leq 17 \] Substituting \( x = 6 \) and \( y = 7 \): \[ 6 + 7 = 13 \leq 17 \] This inequality also holds true.

Step 3: Check Inequality 3

Now, we evaluate the third inequality: \[ 2x + 6y \leq 66 \] Substituting \( x = 6 \) and \( y = 7 \): \[ 2(6) + 6(7) = 12 + 42 = 54 \leq 66 \] This inequality holds true as well.

Step 4: Check Non-negativity Constraints

We check the non-negativity constraints: \[ x \geq 0 \quad \text{and} \quad y \geq 0 \] Substituting \( x = 6 \) and \( y = 7 \): \[ 6 \geq 0 \quad \text{and} \quad 7 \geq 0 \] Both conditions are satisfied.