To solve the given matrix equation using inverse matrices, let's go through the following steps:
[tex]\[ \left[\begin{array}{cc} 4 & 2 \\ -4 & 2 \end{array}\right] \cdot \left[\begin{array}{l} x \\ y \end{array}\right] = \left[\begin{array}{c} 6 \\ 14 \end{array}\right] \][/tex]
### Step 1: Identify the Coefficient Matrix and Constant Matrix
Given matrices:
[tex]\[ A = \left[\begin{array}{cc} 4 & 2 \\ -4 & 2 \end{array}\right] \][/tex]
[tex]\[ B = \left[\begin{array}{c} 6 \\ 14 \end{array}\right] \][/tex]
### Step 2: Calculate the Inverse of Matrix A
To solve for [tex]\( \left[\begin{array}{l} x \\ y \end{array}\right] \)[/tex], we need the inverse of matrix [tex]\( A \)[/tex]:
[tex]\[ A^{-1} = \left[\begin{array}{cc} 0.125 & -0.125 \\ 0.25 & 0.25 \end{array}\right] \][/tex]
### Step 3: Multiply the Inverse of Matrix A with Matrix B
Now, we need to find the product of [tex]\( A^{-1} \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ \left[\begin{array}{cc} 0.125 & -0.125 \\ 0.25 & 0.25 \end{array}\right] \cdot \left[\begin{array}{c} 6 \\ 14 \end{array}\right] = \left[\begin{array}{c} x \\ y \end{array}\right] \][/tex]
### Step 4: Perform the Multiplication
Performing the multiplication:
[tex]\[ x = 0.125 \cdot 6 + (-0.125) \cdot 14 = 0.75 - 1.75 = -1.0 \][/tex]
[tex]\[ y = 0.25 \cdot 6 + 0.25 \cdot 14 = 1.5 + 3.5 = 5.0 \][/tex]
So the solution is:
[tex]\[ \left[\begin{array}{l} x \\ y \end{array}\right] = \left[\begin{array}{c} -1.0 \\ 5.0 \end{array}\right] \][/tex]
### Step 5: Match the Solution with Given Choices
Given choices:
1. [tex]\( (1, -7) \)[/tex]
2. [tex]\( (4, 5) \)[/tex]
3. [tex]\( (1, -5) \)[/tex]
4. [tex]\( (-1, 5) \)[/tex]
Comparing our result with the choices, we find that [tex]\( (-1, 5) \)[/tex] matches our solution.
### Conclusion
The correct choice is:
[tex]\[ \boxed{4} \][/tex]
