Marcos programs his calculator to evaluate a linear function, but he doesn't say what the function is. When 5 is entered, the calculator displays the value 2. When 15 is entered, the calculator displays the value 6. Which of the following expressions explains what the calculator will display when any number, n, is entered?

Options

A. `2/5 n`
B. `5/2 n`
C. `n - 3`
D. `n - 9`

Answer

To solve the problem, we need to determine the linear function that Marcos programmed into his calculator based on the given inputs and outputs. We know that:

- When \( n = 5 \), the output is \( 2 \).
- When \( n = 15 \), the output is \( 6 \).

Let's denote the linear function as \( f(n) = an + b \), where \( a \) is the slope and \( b \) is the y-intercept.

### Step 1: Set up the equations based on the given points.
From the information provided, we can create two equations:

1. \( f(5) = 2 \) gives us:
\[ 5a + b = 2 \quad \text{(Equation 1)} \]

2. \( f(15) = 6 \) gives us:
\[ 15a + b = 6 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.
To eliminate \( b \), we can subtract Equation 1 from Equation 2:

\[ (15a + b) - (5a + b) = 6 - 2 \]

This simplifies to:

\[ 10a = 4 \]

Now, solving for \( a \):

\[ a = \frac{4}{10} = \frac{2}{5} \]

### Step 3: Substitute \( a \) back to find \( b \).
Now that we have \( a \), we can substitute it back into Equation 1 to find \( b \):

\[ 5\left(\frac{2}{5}\right) + b = 2 \]

This simplifies to:

\[ 2 + b = 2 \]

Thus,

\[ b = 0 \]

### Step 4: Write the linear function.
Now we can write the linear function:

\[ f(n) = \frac{2}{5}n + 0 = \frac{2}{5}n \]

### Step 5: Verify the function with the given inputs.
- For \( n = 5 \):
\[ f(5) = \frac{2}{5} \times 5 = 2 \]
- For \( n = 15 \):
\[ f(15) = \frac{2}{5} \times 15 = 6 \]

Both conditions are satisfied.

### Conclusion:
The expression that explains what the calculator will display when any number \( n \) is entered is:

\[ f(n) = \frac{2n}{5} \]