The system of linear equations in x1, x2, x3, where α, β ∈ R, has...

Steps to Solve

1. Write the Augmented Matrix

The system of linear equations can be represented in augmented matrix form as follows:

$$ \begin{pmatrix} 1 & 1 & 1 & | & 3 \ 0 & -1 & 2 & | & 1 \ \alpha & \beta & 0 & | & 0 \end{pmatrix} $$

2. Row Reduction

Next, we apply row operations to simplify the matrix. Start by making zeros below the first leading 1 (the first column):

• Add $- \alpha R_1 + R_3$ to $R_3$.

This gives us a new third row:

$$ R_3 \rightarrow R_3 - \alpha R_1 $$

3. Updated Augmented Matrix

Now the augmented matrix looks like this:

$$ \begin{pmatrix} 1 & 1 & 1 & | & 3 \ 0 & -1 & 2 & | & 1 \ 0 & \beta - \alpha & -\alpha & | & -3\alpha \end{pmatrix} $$

4. Analyze Conditions for Solutions

For the system to have a unique solution, the last row must not lead to a contradiction. Set the condition for $\beta$ and $\alpha$:

• If $\beta - \alpha \neq 0$, the system has a unique solution.
• If $\beta - \alpha = 0$ leading to a zero row in the third leading position, we need to ensure $-3\alpha \neq 0$, which means $\alpha \neq 0$ for infinite solutions.

This gives us conditions for $\alpha$ and $\beta$.

5. Final Conditions for $\alpha$ and $\beta$

The conditions derived are:

• For a unique solution: $\beta \neq \alpha$
• For infinite solutions: $\beta = \alpha$ and $\alpha \neq 0$.