Which point would be a solution to the system of linear inequalities shown below?

y, is greater than, minus, 4, x, plus, 6, y, is greater than, one third, x, minus, 7

*y*>−4*x*+6*y*>31​*x*−7

Answer

Multiple Choice Answers

left parenthesis, 9, comma, minus, 7, right parenthesis(9,−7)

left parenthesis, minus, 12, comma, minus, 2, right parenthesis(−12,−2)

left parenthesis, 12, comma, 1, right parenthesis(12,1)

left parenthesis, minus, 12, comma, minus, 7, right parenthesis(−12,−7)

To determine which point is a solution to the system of linear inequalities, we need to check each point against the given inequalities:

1. **First inequality:** \( y > -4x + 6 \)
2. **Second inequality:** \( y > \frac{1}{3}x - 7 \)

Let's test each of the given points:

### 1. Point \((9, -7)\):
- **First inequality:** \( -7 > -4(9) + 6 \) simplifies to \( -7 > -36 + 6 \) or \( -7 > -30 \) (True)
- **Second inequality:** \( -7 > \frac{1}{3}(9) - 7 \) simplifies to \( -7 > 3 - 7 \) or \( -7 > -4 \) (False)

This point does not satisfy the second inequality, so it is not a solution.

### 2. Point \((-12, -2)\):
- **First inequality:** \( -2 > -4(-12) + 6 \) simplifies to \( -2 > 48 + 6 \) or \( -2 > 54 \) (False)
- **Second inequality:** \( -2 > \frac{1}{3}(-12) - 7 \) simplifies to \( -2 > -4 - 7 \) or \( -2 > -11 \) (True)

This point does not satisfy the first inequality, so it is not a solution.

### 3. Point \((12, 1)\):
- **First inequality:** \( 1 > -4(12) + 6 \) simplifies to \( 1 > -48 + 6 \) or \( 1 > -42 \) (True)
- **Second inequality:** \( 1 > \frac{1}{3}(12) - 7 \) simplifies to \( 1 > 4 - 7 \) or \( 1 > -3 \) (True)

This point satisfies both inequalities.

### 4. Point \((-12, -7)\):
- **First inequality:** \( -7 > -4(-12) + 6 \) simplifies to \( -7 > 48 + 6 \) or \( -7 > 54 \) (False)
- **Second inequality:** \( -7 > \frac{1}{3}(-12) - 7 \) simplifies to \( -7 > -4 - 7 \) or \( -7 > -11 \) (True)

This point does not satisfy the first inequality, so it is not a solution.

### Conclusion:
The only point that satisfies both inequalities is **\((12, 1)\)**.

**Answer: \((12, 1)\)**

Would you like more details or have any questions about this? 

Here are five related questions to consider:

1. How do you graph the boundary lines for linear inequalities?
2. What does the solution region of a system of linear inequalities represent?
3. How can you check if a point is on the boundary line of an inequality?
4. What happens to the solution region when you change the inequality sign (e.g., from \(>\) to \(\geq\))?
5. How do you solve a system of linear inequalities with more than two inequalities?

**Tip:** When checking if a point satisfies an inequality, substitute the point into the inequality and simplify. If the resulting statement is true for both inequalities, the point is a solution to the system.

Discover how to find a solution to a system of linear inequalities step-by-step. Learn which point satisfies both inequalities y > -4x + 6 and y > 1/3x - 7. Suitable for high school students studying linear inequalities in algebra.

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