If `[[5, -4],[-5, 8]][[x],[y]] = [[40],[80]]`, then the values of x and y, respectively, are :

Options

A. 32, 35
B. 30, 35
C. 35, 32
D. 32, 30

Answer

To solve the equation given by the matrix multiplication \(\begin{bmatrix} 5 & -4 \\ -5 & 8 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 40 \\ 80 \end{bmatrix}\), we will follow these steps:

### Step 1: Set up the equations from the matrix multiplication

From the matrix multiplication, we can derive two equations:

1. \(5x - 4y = 40\) (Equation 1)
2. \(-5x + 8y = 80\) (Equation 2)

### Step 2: Solve the equations simultaneously

We can solve these two equations simultaneously. Let's first add Equation 1 and Equation 2.

Adding the two equations:

\[ (5x - 4y) + (-5x + 8y) = 40 + 80 \]

This simplifies to:

\[ 0 + 4y = 120 \]

### Step 3: Solve for \(y\)

Now, we can solve for \(y\):

\[ 4y = 120 \]

Dividing both sides by 4:

\[ y = \frac{120}{4} = 30 \]

### Step 4: Substitute \(y\) back to find \(x\)

Now that we have \(y = 30\), we can substitute this value back into Equation 1 to find \(x\):

Substituting \(y\) into Equation 1:

\[ 5x - 4(30) = 40 \]

This simplifies to:

\[ 5x - 120 = 40 \]

### Step 5: Solve for \(x\)

Now, we can solve for \(x\):

\[ 5x = 40 + 120 \]

\[ 5x = 160 \]

Dividing both sides by 5:

\[ x = \frac{160}{5} = 32 \]

### Final Answer

Thus, the values of \(x\) and \(y\) are:

\[ x = 32, \quad y = 30 \]