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We are given a system of linear equations represented by the augmented matrix:

\[
\begin{bmatrix}
3 & -2 & | & -3 \\
3 & -a & | & 4b
\end{bmatrix}
\]

We are asked to determine the value of the parameter \( a \in \mathbb{R} \) for which the system has no solutions, given that \( b \neq -\frac{3}{4} \).

### Solution Overview:
The system will have no solutions if the system is inconsistent, meaning the two equations are parallel but not coincident. To explore this, we'll perform row reduction and analyze the resulting system.

### Step 1: Row Reduction
We can reduce the system by eliminating one variable to analyze the conditions for inconsistency. Start by subtracting row 1 from row 2:

\[
R_2 \rightarrow R_2 - R_1
\]

This gives us:

\[
\begin{bmatrix}
3 & -2 & | & -3 \\
0 & -a + 2 & | & 4b + 3
\end{bmatrix}
\]

### Step 2: Check for Inconsistency
The system will be inconsistent if we have a row of the form:

\[
0 \quad 0 \quad | \quad c
\]

where \( c \neq 0 \). For this to happen, we need the second equation to have zero coefficients for the variables but a non-zero constant. So, we need:

\[
-a + 2 = 0 \quad \text{and} \quad 4b + 3 \neq 0
\]

Solving for \( a \):

\[
-a + 2 = 0 \quad \Rightarrow \quad a = 2
\]

Now, check the condition for \( b \):

\[
4b + 3 \neq 0 \quad \Rightarrow \quad b \neq -\frac{3}{4}
\]

This confirms that for \( a = 2 \), the system will have no solutions as long as \( b \neq -\frac{3}{4} \), which is already given.

### Final Answer:
The system has no solutions when \( a = 2 \).

---

Would you like further details or have any questions about this solution? Here are five related questions to explore:

1. What conditions guarantee a system of linear equations has infinitely many solutions?
2. How can you determine if a system of equations is inconsistent using Gaussian elimination?
3. What is the geometric interpretation of a system with no solutions in two variables?
4. How does the rank of a matrix relate to the existence of solutions in a linear system?
5. Can you provide another example of a system where parameter values lead to no solutions?

**Tip:** Inconsistent systems arise when the two equations represent parallel lines that never intersect. In matrix terms, this happens when one row becomes zero in the variables but has a non-zero constant.

Consider the system of linear equations represented by the augmented matrix: [[3, -2, -3], [3, -a, 4b]]. For which value of the parameter a ∈ ℝ does the system have no solutions, given that b ≠ -3/4?

Explore how to determine the value of 'a' in a system of linear equations represented by the augmented matrix [[3, -2, -3], [3, -a, 4b]] for which the system has no solutions. Includes step-by-step row reduction and analysis of conditions for inconsistency. Suitable for university-level students.

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