Let `f` be a linear function such that `f(5)=-2` and `f(11)=28`. What is the value of `(f(9)-f(7))/(2)` ?

Answer

To solve the problem, we need to find the value of \((f(9) - f(7)) / 2\) given that \(f\) is a linear function and we know two points on this function: \(f(5) = -2\) and \(f(11) = 28\).

### Step 1: Define the linear function
A linear function can be expressed in the form:
\[ f(x) = ax + b \]
where \(a\) is the slope and \(b\) is the y-intercept.

### Step 2: Set up equations using the given points
Using the information provided:
1. From \(f(5) = -2\):
\[ 5a + b = -2 \quad \text{(Equation 1)} \]
2. From \(f(11) = 28\):
\[ 11a + b = 28 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
We can solve these two equations simultaneously. We will subtract Equation 1 from Equation 2:
\[ (11a + b) - (5a + b) = 28 - (-2) \]
This simplifies to:
\[ 11a - 5a + b - b = 28 + 2 \]
\[ 6a = 30 \]
Now, divide both sides by 6:
\[ a = 5 \]

### Step 4: Substitute \(a\) back to find \(b\)
Now that we have \(a\), we can substitute it back into either Equation 1 or Equation 2 to find \(b\). Using Equation 1:
\[ 5(5) + b = -2 \]
\[ 25 + b = -2 \]
Subtract 25 from both sides:
\[ b = -27 \]

### Step 5: Write the function
Now we have both \(a\) and \(b\):
\[ f(x) = 5x - 27 \]

### Step 6: Calculate \(f(9)\) and \(f(7)\)
Now we can find \(f(9)\) and \(f(7)\):
1. Calculate \(f(9)\):
\[ f(9) = 5(9) - 27 = 45 - 27 = 18 \]
2. Calculate \(f(7)\):
\[ f(7) = 5(7) - 27 = 35 - 27 = 8 \]

### Step 7: Find \((f(9) - f(7)) / 2\)
Now we can find the value of \((f(9) - f(7)) / 2\):
\[ f(9) - f(7) = 18 - 8 = 10 \]
Now divide by 2:
\[ \frac{f(9) - f(7)}{2} = \frac{10}{2} = 5 \]

### Final Answer
The value of \((f(9) - f(7)) / 2\) is \(5\).