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

y, is greater than or equal to, 2, x, plus, 5, y, is greater than or equal to, minus, start fraction, 2, divided by, 5, end fraction, x, plus, 7

*y*≥2*x*+5*y*≥−52​*x*+7

Answer

Multiple Choice Answers

left parenthesis, 0, comma, 8, right parenthesis(0,8)

left parenthesis, 5, comma, 5, right parenthesis(5,5)

left parenthesis, 5, comma, 7, right parenthesis(5,7)

left parenthesis, minus, 10, comma, 10, right parenthesis(−10,10)

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:** \( y \geq 2x + 5 \)
2. **Second inequality:** \( y \geq -\frac{2}{5}x + 7 \)

Let's test each of the given points:

### 1. Point \((0, 8)\):
- **First inequality:** \( 8 \geq 2(0) + 5 \) simplifies to \( 8 \geq 5 \) (True)
- **Second inequality:** \( 8 \geq -\frac{2}{5}(0) + 7 \) simplifies to \( 8 \geq 7 \) (True)

This point satisfies both inequalities.

### 2. Point \((5, 5)\):
- **First inequality:** \( 5 \geq 2(5) + 5 \) simplifies to \( 5 \geq 15 \) (False)
- **Second inequality:** \( 5 \geq -\frac{2}{5}(5) + 7 \) simplifies to \( 5 \geq 5 \) (True)

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

### 3. Point \((5, 7)\):
- **First inequality:** \( 7 \geq 2(5) + 5 \) simplifies to \( 7 \geq 15 \) (False)
- **Second inequality:** \( 7 \geq -\frac{2}{5}(5) + 7 \) simplifies to \( 7 \geq 5 \) (True)

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

### 4. Point \((-10, 10)\):
- **First inequality:** \( 10 \geq 2(-10) + 5 \) simplifies to \( 10 \geq -15 \) (True)
- **Second inequality:** \( 10 \geq -\frac{2}{5}(-10) + 7 \) simplifies to \( 10 \geq 11 \) (False)

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

### Conclusion:
The only point that satisfies both inequalities is **\((0, 8)\)**.

**Answer: \((0, 8)\)**

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

Here are five related questions to consider:

1. How do you graph the solution set for a system of linear inequalities?
2. What is the difference between a strict inequality and a non-strict inequality in terms of solutions?
3. Can a system of linear inequalities have no solution? If so, how can you tell?
4. How do the slopes of the lines in a system of linear inequalities affect the solution region?
5. How can you determine the vertex points of the feasible region in a system of linear inequalities?

**Tip:** When solving systems of inequalities, always check if the point satisfies *both* inequalities to ensure it lies within the solution region.

Learn how to identify solutions to a system of linear inequalities step-by-step. This problem involves testing points against linear inequalities and determining which points satisfy both inequalities. Suitable for Grades 7-9 students.

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