Finding x and y

Solution:

Step 1: Expand the equation

\[ \begin{bmatrix} 2x \\ x \end{bmatrix} + \begin{bmatrix} 3y \\ 5y \end{bmatrix} + \begin{bmatrix} -8 \\ 11 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]

Step 2: Combine terms

\[ \begin{bmatrix} 2x + 3y – 8 \\ x + 5y + 11 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]

Step 3: Form equations

\[ 2x + 3y – 8 = 0 \Rightarrow 2x + 3y = 8 \quad …(1) \] \[ x + 5y + 11 = 0 \Rightarrow x + 5y = -11 \quad …(2) \]

Step 4: Solve equations

From (2): \[ x = -11 – 5y \] Substitute into (1): \[ 2(-11 – 5y) + 3y = 8 \] \[ -22 – 10y + 3y = 8 \Rightarrow -22 – 7y = 8 \Rightarrow -7y = 30 \Rightarrow y = -\frac{30}{7} \] \[ x = -11 – 5\left(-\frac{30}{7}\right) = -11 + \frac{150}{7} = \frac{-77 + 150}{7} = \frac{73}{7} \]

Final Answer:

\[ x = \frac{73}{7}, \quad y = -\frac{30}{7} \]