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

Options

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

Explanation

To solve the problem, we start with the function given as \( f(x) = ax + b \), where \( a \) and \( b \) are integers. We are provided with two conditions: \( f(-1) = -5 \) and \( f(3) = 3 \). We will use these conditions to form equations and solve for \( a \) and \( b \).

### Step-by-step Solution:

1. **Substituting the first condition**:
We substitute \( x = -1 \) into the function:
\[ f(-1) = a(-1) + b = -a + b \]
According to the problem, \( f(-1) = -5 \). Therefore, we have:
\[ -a + b = -5 \quad \text{(Equation 1)} \]

2. **Substituting the second condition**:
Now we substitute \( x = 3 \) into the function:
\[ f(3) = a(3) + b = 3a + b \]
According to the problem, \( f(3) = 3 \). Therefore, we have:
\[ 3a + b = 3 \quad \text{(Equation 2)} \]

3. **Solving the equations**:
We now have two equations:
- Equation 1: \( -a + b = -5 \)
- Equation 2: \( 3a + b = 3 \)

We can eliminate \( b \) by subtracting Equation 1 from Equation 2:
\[ (3a + b) - (-a + b) = 3 - (-5) \]
Simplifying this gives:
\[ 3a + b + a - b = 3 + 5 \]
\[ 4a = 8 \]
Dividing both sides by 4:
\[ a = 2 \]

4. **Finding \( b \)**:
Now that we have \( a \), we can substitute it back into Equation 1 to find \( b \):
\[ -a + b = -5 \]
Substituting \( a = 2 \):
\[ -2 + b = -5 \]
Adding 2 to both sides:
\[ b = -5 + 2 = -3 \]

5. **Final values**:
Thus, the values of \( a \) and \( b \) are:
\[ a = 2, \quad b = -3 \]

### Conclusion:
The final answer is:
\[ a = 2, \quad b = -3 \]