Find `[(x),(y)]if[(3,2),(-1,4)][(x),(y)]=[(-5),(4)]`.

Options

A. `[(-2,0.5)]`
B. `[(-5//6),(1)]`
C. `[(-1,3//4)]`
D. `[(-2),(1//2)]`

Explanation

To solve the equation \(\begin{pmatrix} 3 & 2 \\ -1 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -5 \\ 4 \end{pmatrix}\), we will follow these steps:

### Step 1: Set up the equations from the matrix multiplication
From the multiplication of the matrices, we can derive two equations:

1. \(3x + 2y = -5\) (from the first row)
2. \(-x + 4y = 4\) (from the second row)

### Step 2: Rearrange the second equation
We can rearrange the second equation to express \(x\) in terms of \(y\):

\[ -x + 4y = 4 \implies -x = 4 - 4y \implies x = 4y - 4 \]

### Step 3: Substitute \(x\) in the first equation
Now, substitute \(x\) in the first equation:

\[ 3(4y - 4) + 2y = -5 \]

### Step 4: Simplify the equation
Expanding and simplifying the equation:

\[ 12y - 12 + 2y = -5 \]
\[ 14y - 12 = -5 \]

### Step 5: Solve for \(y\)
Now, add 12 to both sides:

\[ 14y = 7 \]
\[ y = \frac{7}{14} = \frac{1}{2} \]

### Step 6: Substitute \(y\) back to find \(x\)
Now substitute \(y = \frac{1}{2}\) back into the equation we derived for \(x\):

\[ x = 4\left(\frac{1}{2}\right) - 4 \]
\[ x = 2 - 4 = -2 \]

### Final Answer
Thus, the values are:
\[ x = -2, \quad y = \frac{1}{2} \]