`if x[{:(2),(3):}]+y[{:(-1),(5):}]=[{:(4),(19):}],`then the values of `x` and `y` are:

Options

A. `x=y,y=3`
B. `x=6,y=8`
C. `x=3,y=2`
D. None of these

Explanation

To solve the equation \( x \begin{pmatrix} 2 \\ 3 \end{pmatrix} + y \begin{pmatrix} -1 \\ 5 \end{pmatrix} = \begin{pmatrix} 4 \\ 19 \end{pmatrix} \), we will follow these steps:

### Step 1: Set up the equations
We can express the equation in terms of its components:
1. From the first row: \( 2x - y = 4 \) (Equation 1)
2. From the second row: \( 3x + 5y = 19 \) (Equation 2)

### Step 2: Solve Equation 1 for \( y \)
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 2x - 4 \]

### Step 3: Substitute \( y \) in Equation 2
Now, we substitute \( y \) in Equation 2:
\[ 3x + 5(2x - 4) = 19 \]
Expanding this gives:
\[ 3x + 10x - 20 = 19 \]
Combining like terms:
\[ 13x - 20 = 19 \]

### Step 4: Solve for \( x \)
Now, we isolate \( x \):
\[ 13x = 19 + 20 \]
\[ 13x = 39 \]
\[ x = \frac{39}{13} = 3 \]

### Step 5: Substitute \( x \) back to find \( y \)
Now that we have \( x \), we can find \( y \) using the expression we derived in Step 2:
\[ y = 2(3) - 4 \]
\[ y = 6 - 4 = 2 \]

### Final Answer
Thus, the values of \( x \) and \( y \) are:
\[ x = 3, \quad y = 2 \]

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