The problem asks you to determine the values of $x$ and $y$ from matrix equations. Let's go through each part one by one.

Part a:

$\begin{pmatrix} x \\ 5 \end{pmatrix} = \begin{pmatrix} 7 \\ 2y \end{pmatrix} + \begin{pmatrix} -1 \\ -11 \end{pmatrix}$

First, let's simplify the right-hand side: $\begin{pmatrix} 7 \\ 2y \end{pmatrix} + \begin{pmatrix} -1 \\ -11 \end{pmatrix} = \begin{pmatrix} 7 - 1 \\ 2y - 11 \end{pmatrix} = \begin{pmatrix} 6 \\ 2y - 11 \end{pmatrix}$

Now we have: $\begin{pmatrix} x \\ 5 \end{pmatrix} = \begin{pmatrix} 6 \\ 2y - 11 \end{pmatrix}$

By comparing elements of the vectors:

1. $x = 6$
2. $5 = 2y - 11$

Solving for $y$ in the second equation: $5 + 11 = 2y \implies 16 = 2y \implies y = 8$

Thus, for part (a), the solution is: $x = 6, \quad y = 8$

Part b:

$\begin{pmatrix} 9 \\ 2y \end{pmatrix} + \begin{pmatrix} -6 \\ -y \end{pmatrix} = \begin{pmatrix} 3 \\ y \end{pmatrix}$

First, simplify the left-hand side: $\begin{pmatrix} 9 \\ 2y \end{pmatrix} + \begin{pmatrix} -6 \\ -y \end{pmatrix} = \begin{pmatrix} 9 - 6 \\ 2y - y \end{pmatrix} = \begin{pmatrix} 3 \\ y \end{pmatrix}$

Now we have: $\begin{pmatrix} 3 \\ y \end{pmatrix} = \begin{pmatrix} 3 \\ y \end{pmatrix}$

This is an identity, meaning that part (b) holds for any value of $y$. Therefore, $y$ is undetermined, and any $y$ satisfies the equation.

Part c:

$\begin{pmatrix} 2x \\ 2y \end{pmatrix} + \begin{pmatrix} 3y \\ 3x \end{pmatrix} = \begin{pmatrix} 11 \\ 4 \end{pmatrix}$

First, simplify the left-hand side: $\begin{pmatrix} 2x \\ 2y \end{pmatrix} + \begin{pmatrix} 3y \\ 3x \end{pmatrix} = \begin{pmatrix} 2x + 3y \\ 2y + 3x \end{pmatrix}$

Now we have the equation: $\begin{pmatrix} 2x + 3y \\ 2y + 3x \end{pmatrix} = \begin{pmatrix} 11 \\ 4 \end{pmatrix}$

By comparing elements of the vectors:

1. $2x + 3y = 11$
2. $2y + 3x = 4$

Now we have a system of two equations: $2x + 3y = 11 \quad \text{(Equation 1)}$ $2y + 3x = 4 \quad \text{(Equation 2)}$

Let me solve this system.

I'll subtract Equation 1 from Equation 2: $(2y + 3x) - (2x + 3y) = 4 - 11$ $2y + 3x - 2x - 3y = -7$ $(x - y) = -7$ Thus: $x = y - 7 \quad \text{(Equation 3)}$

Now substitute $x = y - 7$ into Equation 1: $2(y - 7) + 3y = 11$ $2y - 14 + 3y = 11$ $5y = 25$ $y = 5$

Substitute $y = 5$ into Equation 3: $x = 5 - 7 = -2$

Thus, for part (c), the solution is: $x = -2, \quad y = 5$

Summary of Solutions:

• Part a: $x = 6, y = 8$
• Part b: $y$ is undetermined.
• Part c: $x = -2, y = 5$

Would you like more detailed explanations or have any questions?

Here are 5 related questions to further practice:

1. How would you solve a system of linear equations using matrix inversion?
2. Can you express matrix addition geometrically or graphically?
3. What is the determinant of a matrix, and how does it relate to the solutions of linear systems?
4. How do you determine if a system of equations has one solution, no solution, or infinite solutions?
5. What is the method of Gaussian elimination, and how does it help solve matrix equations?

Tip: When solving systems of equations, always look for symmetries or patterns that can simplify calculations before diving into complex algebraic manipulations.