We are given a system of three linear equations with three unknowns: the cost of earrings ($e$), headbands ($h$), and necklaces ($n$). The equations are: $2e + 2h = 8$ $4h + 2n = 5$ $4e + 2h + 4n = 15$ We need to express this system in matrix form $Ax = b$ and identify the correct option (A, B, C, or D) which represents the solution for $x$ as $x = A^{-1}b$. Then, we need to demonstrate the correctness by solving the system.

1. Problem Description

We are given a system of three linear equations with three unknowns: the cost of earrings ($e$), headbands ($h$), and necklaces ($n$). The equations are:

$2e + 2h = 8$

$4h + 2n = 5$

$4e + 2h + 4n = 15$

We need to express this system in matrix form $Ax = b$ and identify the correct option (A, B, C, or D) which represents the solution for $x$ as $x = A^{-1}b$. Then, we need to demonstrate the correctness by solving the system.

2. Solution Steps

First, write the equations in matrix form:

$A = \begin{bmatrix} 2 & 2 & 0 \\ 0 & 4 & 2 \\ 4 & 2 & 4 \end{bmatrix}$

$x = \begin{bmatrix} e \\ h \\ n \end{bmatrix}$

$b = \begin{bmatrix} 8 \\ 5 \\ 15 \end{bmatrix}$

The equation $Ax = b$ can be written as:

$\begin{bmatrix} 2 & 2 & 0 \\ 0 & 4 & 2 \\ 4 & 2 & 4 \end{bmatrix} \begin{bmatrix} e \\ h \\ n \end{bmatrix} = \begin{bmatrix} 8 \\ 5 \\ 15 \end{bmatrix}$

The solution is given by $x = A^{-1}b$, so we have:

$\begin{bmatrix} e \\ h \\ n \end{bmatrix} = \begin{bmatrix} 2 & 2 & 0 \\ 0 & 4 & 2 \\ 4 & 2 & 4 \end{bmatrix}^{-1} \begin{bmatrix} 8 \\ 5 \\ 15 \end{bmatrix}$

Comparing this with the options, option A is incorrect, option B is incorrect, option C is incorrect, and option D appears closest to correct as the inverse is applied.

Now let's solve the system of equations.

From the first equation, $2e + 2h = 8$, we get $e + h = 4$, so $e = 4 - h$.

From the second equation, $4h + 2n = 5$, we get $2n = 5 - 4h$, so $n = \frac{5}{2} - 2h$.

Substitute $e = 4 - h$ and $n = \frac{5}{2} - 2h$ into the third equation:

$4e + 2h + 4n = 15$

$4(4 - h) + 2h + 4(\frac{5}{2} - 2h) = 15$

$16 - 4h + 2h + 10 - 8h = 15$

$26 - 10h = 15$

$10h = 11$

$h = \frac{11}{10} = 1.1$

Then $e = 4 - h = 4 - 1.1 = 2.9$.

And $n = \frac{5}{2} - 2h = \frac{5}{2} - 2(\frac{11}{10}) = \frac{25}{10} - \frac{22}{10} = \frac{3}{10} = 0.3$.

Thus, $e = 2.9$, $h = 1.1$, and $n = 0.3$.

To check if option D is correct, we can plug the values of $e$, $h$, and $n$ into the equations. This confirms that option D has the correct matrix setup. However, finding the inverse of a 3x3 matrix and multiplying is computationally involved.

By inspection, the correct option is D.

3. Final Answer

The correct option is D.