Question 2 · Álgebra UOC Juny 2023

Question 2 (25% of final grade). Linear systems with parameters

Consider the parameter $a \in \{0, 1, \dots, 9\}$ and the following system of linear equations in three variables $x, y, z$:$$\begin{cases}x + y + z = 1 \\ (a + 1)x + ky + (a + 1)z = a + 1 \\ (a + 1)x + (a + 1)y + kz = a + 1 \\ (a + 1)x + (a + 1)y + (a + 1)z = k\end{cases}$$

a.

Classify the system according to the values of the real parameter $k \in \mathbb{R}$.

b.

Discard the fourth equation to obtain a square system of three equations in three unknowns. Find its general solution for the particular case $k = 2023$.