`f(x)=(x-k)/(5) and g(x)=5x+7`. If `f(g(x))=g(f(x))`, what is the value of k?

Options

A. 2
B. 4
C. 5
D. 7

Answer

To solve the equation \( f(g(x)) = g(f(x)) \) where \( f(x) = \frac{x - k}{5} \) and \( g(x) = 5x + 7 \), we will follow these steps:

### Step 1: Calculate \( f(g(x)) \)

First, we need to find \( g(x) \):
\[ g(x) = 5x + 7 \]

Now, substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(5x + 7) = \frac{(5x + 7) - k}{5} \]

### Step 2: Simplify \( f(g(x)) \)

Now simplify the expression:
\[ f(g(x)) = \frac{5x + 7 - k}{5} = x + \frac{7 - k}{5} \]

### Step 3: Calculate \( g(f(x)) \)

Next, we need to find \( f(x) \):
\[ f(x) = \frac{x - k}{5} \]

Now, substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g\left(\frac{x - k}{5}\right) = 5\left(\frac{x - k}{5}\right) + 7 \]

### Step 4: Simplify \( g(f(x)) \)

Now simplify the expression:
\[ g(f(x)) = (x - k) + 7 = x - k + 7 = x + (7 - k) \]

### Step 5: Set \( f(g(x)) \) equal to \( g(f(x)) \)

Now we have:
\[ f(g(x)) = x + \frac{7 - k}{5} \]
\[ g(f(x)) = x + (7 - k) \]

Set them equal to each other:
\[ x + \frac{7 - k}{5} = x + (7 - k) \]

### Step 6: Eliminate \( x \) and solve for \( k \)

Since \( x \) appears on both sides, we can eliminate it:
\[ \frac{7 - k}{5} = 7 - k \]

### Step 7: Multiply both sides by 5 to eliminate the fraction

Multiply both sides by 5:
\[ 7 - k = 5(7 - k) \]

### Step 8: Distribute on the right side

Distributing gives:
\[ 7 - k = 35 - 5k \]

### Step 9: Rearrange the equation

Rearranging the equation:
\[ 5k - k = 35 - 7 \]
\[ 4k = 28 \]

### Step 10: Solve for \( k \)

Divide both sides by 4:
\[ k = 7 \]

### Final Answer:

The value of \( k \) is \( \boxed{7} \).

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