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This exercise requires solving systems of linear equations using Cramer's rule, which involves calculating determinants. I'll solve system A as an example. The system A is:

\[
\begin{cases}
2x - 2y = 2 \\
3x + 5y = -16
\end{cases}
\]

### Cramer's Rule Steps:
1. Write the system in matrix form:
   \[
   \begin{pmatrix}
   2 & -2 \\
   3 & 5
   \end{pmatrix}
   \begin{pmatrix}
   x \\
   y
   \end{pmatrix}
   =
   \begin{pmatrix}
   2 \\
   -16
   \end{pmatrix}
   \]

2. Calculate the determinant of the coefficient matrix \(D\):
   \[
   D = \text{det} \begin{pmatrix}
   2 & -2 \\
   3 & 5
   \end{pmatrix}
   = (2)(5) - (-2)(3) = 10 + 6 = 16
   \]

3. To find \(x\), replace the first column of the coefficient matrix with the constants (right-hand side of the equations), and calculate the determinant \(D_x\):
   \[
   D_x = \text{det} \begin{pmatrix}
   2 & -2 \\
   -16 & 5
   \end{pmatrix}
   = (2)(5) - (-2)(-16) = 10 - 32 = -22
   \]

   Therefore, \(x = \frac{D_x}{D} = \frac{-22}{16} = -\frac{11}{8}\).

4. To find \(y\), replace the second column of the coefficient matrix with the constants, and calculate the determinant \(D_y\):
   \[
   D_y = \text{det} \begin{pmatrix}
   2 & 2 \\
   3 & -16
   \end{pmatrix}
   = (2)(-16) - (2)(3) = -32 - 6 = -38
   \]

   Therefore, \(y = \frac{D_y}{D} = \frac{-38}{16} = -\frac{19}{8}\).

### Final solution for system A:
\[
x = -\frac{11}{8}, \quad y = -\frac{19}{8}
\]

---

Would you like me to solve another system from the exercise, or provide more details on any step?

Here are 5 related questions:
1. How do you calculate the determinant of a 2x2 matrix?
2. Can Cramer's rule be applied to any system of linear equations?
3. What happens if the determinant of the coefficient matrix is zero?
4. How would you solve a 3x3 system using Cramer's rule?
5. What are some alternative methods to solve systems of equations?

**Tip:** Always check if the determinant is non-zero before applying Cramer's rule. If it is zero, the system may have no solution or infinitely many solutions.

Resolve the following systems of equations by Cramer's rule.

Explore how to solve systems of equations using Cramer's Rule, including the calculation of determinants and application of formulas for variables. This method is demonstrated with detailed steps and examples, suitable for Grade 9 Algebra.

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