which of the following point lies in solution region of inequality `3x-yle5`?

Options

A. (5,1)
B. (1,5)
C. (2,0)
D. (2,-1)

Answer

To determine which of the given points lies in the solution region of the inequality \(3x - y \leq 5\), we will substitute each point into the inequality and check if it satisfies the condition.

### Step-by-Step Solution:

1. **Identify the inequality**:
The inequality we need to check is \(3x - y \leq 5\).

2. **Check the first point (5, 1)**:
- Substitute \(x = 5\) and \(y = 1\) into the inequality:
\[ 3(5) - 1 \leq 5 \]
- Calculate:
\[ 15 - 1 = 14 \]
- Check if \(14 \leq 5\): This is **false**. Therefore, the point (5, 1) does not satisfy the inequality.

3. **Check the second point (1, 5)**:
- Substitute \(x = 1\) and \(y = 5\) into the inequality:
\[ 3(1) - 5 \leq 5 \]
- Calculate:
\[ 3 - 5 = -2 \]
- Check if \(-2 \leq 5\): This is **true**. Therefore, the point (1, 5) satisfies the inequality.

4. **Check the third point (2, 0)**:
- Substitute \(x = 2\) and \(y = 0\) into the inequality:
\[ 3(2) - 0 \leq 5 \]
- Calculate:
\[ 6 - 0 = 6 \]
- Check if \(6 \leq 5\): This is **false**. Therefore, the point (2, 0) does not satisfy the inequality.

5. **Check the fourth point (2, -1)**:
- Substitute \(x = 2\) and \(y = -1\) into the inequality:
\[ 3(2) - (-1) \leq 5 \]
- Calculate:
\[ 6 + 1 = 7 \]
- Check if \(7 \leq 5\): This is **false**. Therefore, the point (2, -1) does not satisfy the inequality.

### Conclusion:
The only point that lies in the solution region of the inequality \(3x - y \leq 5\) is **(1, 5)**.