If f(x) = ax + b, where a and b are integers, f(-1) = -5 and f(3) = 3, then a and b are respectively

Options

A. a = -3, -b = -1
B. a = 2, b= 3
C. a = 2, b = -3
D. a = 0, b = 2

Answer

To solve the problem, we start with the function given by \( f(x) = ax + b \), where \( a \) and \( b \) are integers. We are provided with two conditions: \( f(-1) = -5 \) and \( f(3) = 3 \).

### Step-by-step Solution:

1. **Set up the equations based on the given conditions:**
- From \( f(-1) = -5 \):
\[ f(-1) = a(-1) + b = -5 \implies -a + b = -5 \quad \text{(Equation 1)} \]
- From \( f(3) = 3 \):
\[ f(3) = a(3) + b = 3 \implies 3a + b = 3 \quad \text{(Equation 2)} \]

2. **Now we have two equations:**
- Equation 1: \( -a + b = -5 \)
- Equation 2: \( 3a + b = 3 \)

3. **Subtract Equation 1 from Equation 2:**
\[ (3a + b) - (-a + b) = 3 - (-5) \]
This simplifies to:
\[ 3a + b + a - b = 3 + 5 \]
Which further simplifies to:
\[ 4a = 8 \]

4. **Solve for \( a \):**
\[ a = \frac{8}{4} = 2 \]

5. **Substitute \( a \) back into one of the original equations to find \( b \):**
- Using Equation 2:
\[ 3(2) + b = 3 \]
This simplifies to:
\[ 6 + b = 3 \]
Therefore:
\[ b = 3 - 6 = -3 \]

6. **Final values:**
- We have found \( a = 2 \) and \( b = -3 \).

### Conclusion:
Thus, the values of \( a \) and \( b \) are \( 2 \) and \( -3 \) respectively.