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The problem asks to find all values of \( a \) for which the system of equations has no solution.

Let's focus on **Exercise 36**, which involves the following system of linear equations:

\[
1. \quad x_1 + 2x_2 = -3
\]
\[
2. \quad ax_1 - 2x_2 = 5
\]

For this system to have **no solution**, the lines represented by the two equations must be **parallel** but not coincide. This occurs when the ratios of the coefficients of \(x_1\) and \(x_2\) are the same, but the constants on the right-hand side differ.

### Steps:
1. **Rewrite both equations in standard form:**
   - Equation 1: \(x_1 + 2x_2 = -3\)
   - Equation 2: \(ax_1 - 2x_2 = 5\)

2. **Compare the ratios** of the coefficients of \(x_1\) and \(x_2\) in both equations. The system has no solution if the ratios of the coefficients are equal, but the constants are not.

   For no solution:
   \[
   \frac{1}{a} = \frac{2}{-2} \quad \text{and} \quad \text{the constants should not be equal.}
   \]

   This simplifies to:
   \[
   \frac{1}{a} = -1
   \]

   Solving for \( a \):
   \[
   a = -1
   \]

3. **Check the constants**:
   The constants on the right-hand side of the equations are \(-3\) and 5. Since \(-3 \neq 5\), the constants are different, which confirms that the system is **inconsistent** for \(a = -1\). Therefore, the system will have no solution for \(a = -1\).

### Final Answer:
The system has no solution when \(a = -1\).

Would you like further details or clarifications?

Here are some follow-up questions you may want to consider:
1. How do we determine if a system of linear equations has no solution?
2. What does it mean for two lines to be parallel in terms of their slopes?
3. What is the role of comparing the ratios of coefficients in solving systems of equations?
4. Can a system of equations have infinitely many solutions, and under what conditions?
5. What would the solution look like if \( a \) were not equal to \(-1\)?

**Tip:** When checking for no solution in a system of linear equations, always compare both the coefficients of variables and the constants carefully.

In Exercises 36-40, find all values of a for which the system has no solution. For Exercise 36: x1 + 2x2 = -3 and ax1 - 2x2 = 5.

Learn how to find the value of a for which the system of linear equations x1 + 2x2 = -3 and ax1 - 2x2 = 5 has no solution. This exercise involves understanding the conditions for parallel lines and inconsistent systems. Suitable for students in Grades 10-12.

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