Questions: What ordered pair is the solution to the following system of linear inequalities? 4x - 5y ≥ 11 3x + 6y < 1 (4,2) (0,0) (2,-1) (-1,1)

Transcript text: What ordered pair is the solution to the following system of linear inequalities? \[ \begin{array}{l} 4 x-5 y \geq 11 \\ 3 x+6 y<1 \end{array} \] $(4,2)$ $(0,0)$ $(2,-1)$ $(-1,1)$

Solution

Solution Steps

Step 1: Substitute Ordered Pairs

We will substitute each ordered pair into the inequalities to check if they satisfy both conditions. The inequalities are: \[ 4x - 5y \geq 11 \] \[ 3x + 6y < 1 \]

Step 2: Check Pair (4, 2)

Substituting $ (4, 2) $: \[ 4(4) - 5(2) = 16 - 10 = 6 \quad (\text{not } \geq 11) \] \[ 3(4) + 6(2) = 12 + 12 = 24 \quad (\text{not } < 1) \] This pair does not satisfy the inequalities.

Step 3: Check Pair (0, 0)

Substituting $ (0, 0) $: \[ 4(0) - 5(0) = 0 \quad (\text{not } \geq 11) \] \[ 3(0) + 6(0) = 0 \quad (\text{not } < 1) \] This pair does not satisfy the inequalities.

Step 4: Check Pair (2, -1)

Substituting $ (2, -1) $: \[ 4(2) - 5(-1) = 8 + 5 = 13 \quad (\text{satisfies } \geq 11) \] \[ 3(2) + 6(-1) = 6 - 6 = 0 \quad (\text{satisfies } < 1) \] This pair satisfies both inequalities.

Step 5: Check Pair (-1, 1)

Substituting $ (-1, 1) $: \[ 4(-1) - 5(1) = -4 - 5 = -9 \quad (\text{not } \geq 11) \] \[ 3(-1) + 6(1) = -3 + 6 = 3 \quad (\text{not } < 1) \] This pair does not satisfy the inequalities.

Step 6: Conclusion

The ordered pair $ (2, -1) $ is the only pair that satisfies both inequalities in the system.