Question

Consider the following systems of linear inequalities
x+2y>=10
3x-4y>12
Which of the following orders pairs are solutions to the system?
(CHOOSE ALL THAT APPLY)
A.(16,9)
B.(10,2)
C.(8,1)
D.(2,6)
E.(4,-4)
F.(14,-1)
G.(-4,2)

Answer 1

To determine which ordered pairs are solutions to the system of linear inequalities, we will check each pair against both inequalities:

1. **Inequality 1:** \(x + 2y \geq 10\)
2. **Inequality 2:** \(3x - 4y > 12\)

Now, we substitute each ordered pair into the inequalities.

**A. (16, 9)**
- For Inequality 1: \(16 + 2(9) = 16 + 18 = 34 \geq 10\) (True)
- For Inequality 2: \(3(16) - 4(9) = 48 - 36 = 12 > 12\) (False)

**B. (10, 2)**
- For Inequality 1: \(10 + 2(2) = 10 + 4 = 14 \geq 10\) (True)
- For Inequality 2: \(3(10) - 4(2) = 30 - 8 = 22 > 12\) (True)

**C. (8, 1)**
- For Inequality 1: \(8 + 2(1) = 8 + 2 = 10 \geq 10\) (True)
- For Inequality 2: \(3(8) - 4(1) = 24 - 4 = 20 > 12\) (True)

**D. (2, 6)**
- For Inequality 1: \(2 + 2(6) = 2 + 12 = 14 \geq 10\) (True)
- For Inequality 2: \(3(2) - 4(6) = 6 - 24 = -18 > 12\) (False)

**E. (4, -4)**
- For Inequality 1: \(4 + 2(-4) = 4 - 8 = -4 \geq 10\) (False)
- For Inequality 2: \(3(4) - 4(-4) = 12 + 16 = 28 > 12\) (True)

**F. (14, -1)**
- For Inequality 1: \(14 + 2(-1) = 14 - 2 = 12 \geq 10\) (True)
- For Inequality 2: \(3(14) - 4(-1) = 42 + 4 = 46 > 12\) (True)

**G. (-4, 2)**
- For Inequality 1: \(-4 + 2(2) = -4 + 4 = 0 \geq 10\) (False)
- For Inequality 2: \(3(-4) - 4(2) = -12 - 8 = -20 > 12\) (False)

Now, let's summarize which points satisfy both inequalities:

- A: False
- B: True
- C: True
- D: False
- E: False
- F: True
- G: False

The ordered pairs that are solutions to the system of inequalities are:

**B. (10, 2)**
**C. (8, 1)**
**F. (14, -1)**

Therefore, the correct answers are **B, C, and F**.