Let's break down and solve the two questions presented in the image:

Question 2:

We are given the following system of linear equations:

$x - y = 3$ $2x - 2y = k$

We are asked to determine for which values of $k$ the system has:

• No solution
• Exactly one solution
• Infinitely many solutions

Step 1: Analyze the system

The two equations can be rewritten as:

1. $x - y = 3$
2. $2x - 2y = k$

The second equation can be simplified by dividing everything by 2: $x - y = \frac{k}{2}$

So now, the system becomes:

1. $x - y = 3$
2. $x - y = \frac{k}{2}$

Step 2: Check the conditions

• Infinitely many solutions: This happens when the two equations represent the same line. For that to happen, the constants on the right-hand side must be equal. Therefore: $3 = \frac{k}{2}$ Solving for $k$: $k = 6$ So, for $k = 6$, the system has infinitely many solutions.

• No solution: If the two equations are parallel but not the same, there is no solution. This happens when the left-hand sides are identical, but the constants on the right-hand sides are different. For: $x - y = 3 \quad \text{and} \quad x - y = \frac{k}{2}$ There is no solution if $\frac{k}{2} \neq 3$, i.e., when $k \neq 6$.

• Exactly one solution: If $x - y = 3$ and $2x - 2y = k$ are not parallel (different left-hand sides), the system would have exactly one solution. However, in this case, the left-hand sides are always identical, so this case is not possible. Thus, there can never be exactly one solution.

Summary for Question 2:

• No solution for $k \neq 6$.
• Infinitely many solutions for $k = 6$.
• Exactly one solution is not possible for any $k$.

Question 3:

We are given the following system of linear equations: $2x_2 + x_3 = -2$ $3x_1 + 5x_2 - 5x_3 = 1$ $2x_1 + 4x_2 - 2x_3 = 2$

We will solve this system using either substitution, elimination, or matrix methods like Gaussian elimination.

Step 1: Write in matrix form

The system can be written as: