Find x and y if:
(i) `3[(4),(x)] +2[(y),(-3)]=[(10),( 0)]`
(ii) `x [{:(,-1),(,2):}] -4[{:(,-2),(,-y):}]=[{:(,7),(,-8):}]`

Answer

Let's solve the problem step by step.

### Part (i)
We need to solve the equation:
\[ 3 \begin{pmatrix} 4 \\ x \end{pmatrix} + 2 \begin{pmatrix} y \\ -3 \end{pmatrix} = \begin{pmatrix} 10 \\ 0 \end{pmatrix} \]

**Step 1: Multiply the matrices by their respective scalars.**
\[ 3 \begin{pmatrix} 4 \\ x \end{pmatrix} = \begin{pmatrix} 3 \cdot 4 \\ 3 \cdot x \end{pmatrix} = \begin{pmatrix} 12 \\ 3x \end{pmatrix} \]
\[ 2 \begin{pmatrix} y \\ -3 \end{pmatrix} = \begin{pmatrix} 2 \cdot y \\ 2 \cdot -3 \end{pmatrix} = \begin{pmatrix} 2y \\ -6 \end{pmatrix} \]

**Step 2: Add the two resulting matrices.**
\[ \begin{pmatrix} 12 \\ 3x \end{pmatrix} + \begin{pmatrix} 2y \\ -6 \end{pmatrix} = \begin{pmatrix} 12 + 2y \\ 3x - 6 \end{pmatrix} \]

**Step 3: Set the resulting matrix equal to the right-hand side matrix.**
\[ \begin{pmatrix} 12 + 2y \\ 3x - 6 \end{pmatrix} = \begin{pmatrix} 10 \\ 0 \end{pmatrix} \]

**Step 4: Equate the corresponding elements of the matrices.**
1. From the first row:
\[ 12 + 2y = 10 \]
\[ 2y = 10 - 12 \implies 2y = -2 \implies y = -1 \]

2. From the second row:
\[ 3x - 6 = 0 \]
\[ 3x = 6 \implies x = \frac{6}{3} = 2 \]

Thus, the values are:
\[ x = 2, \quad y = -1 \]

### Part (ii)
We need to solve the equation:
\[ x \begin{pmatrix} -1 & 2 \\ -4 & -y \end{pmatrix} - 4 \begin{pmatrix} -2 & -y \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 7 & -8 \end{pmatrix} \]

**Step 1: Multiply the first matrix by x and the second matrix by -4.**
\[ x \begin{pmatrix} -1 & 2 \\ -4 & -y \end{pmatrix} = \begin{pmatrix} -x & 2x \\ -4x & -xy \end{pmatrix} \]
\[ -4 \begin{pmatrix} -2 & -y \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 8 & 4y \\ 0 & 0 \end{pmatrix} \]

**Step 2: Combine the two resulting matrices.**
\[ \begin{pmatrix} -x & 2x \\ -4x & -xy \end{pmatrix} + \begin{pmatrix} 8 & 4y \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} -x + 8 & 2x + 4y \\ -4x & -xy \end{pmatrix} \]

**Step 3: Set the resulting matrix equal to the right-hand side matrix.**
\[ \begin{pmatrix} -x + 8 & 2x + 4y \\ -4x & -xy \end{pmatrix} = \begin{pmatrix} 7 & -8 \\ 0 & 0 \end{pmatrix} \]

**Step 4: Equate the corresponding elements of the matrices.**
1. From the first row, first column:
\[ -x + 8 = 7 \implies -x = 7 - 8 \implies -x = -1 \implies x = 1 \]

2. From the first row, second column:
\[ 2x + 4y = -8 \]
Substitute \( x = 1 \):
\[ 2(1) + 4y = -8 \implies 2 + 4y = -8 \implies 4y = -8 - 2 \implies 4y = -10 \implies y = -\frac{10}{4} = -\frac{5}{2} \]

Thus, the values are:
\[ x = 1, \quad y = -\frac{5}{2} \]

### Summary of Solutions
- For part (i): \( x = 2, y = -1 \)
- For part (ii): \( x = 1, y = -\frac{5}{2} \)