https://file.aiquickdraw.com/mathbot/file/17297725942842ezohfs4.jpg

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:

\[
\begin{pmatrix}
0 & 2 & 1 \\
3 & 5 & -5 \\
2 & 4 & -2
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
-2 \\
1 \\
2
\end{pmatrix}
\]

#### Step 2: Gaussian elimination
Perform row operations to reduce the matrix to upper triangular form and solve using back substitution.

Let's perform these steps:

- First, leave the first row as is. Multiply the first row by 1.5 and subtract from the second row, then multiply by 1 and subtract from the third row.
- Continue row reduction until you get an upper triangular matrix.

#### Step 3: Back substitution
Once we have an upper triangular matrix, solve for \(x_3\), then \(x_2\), and finally \(x_1\).

By solving this, we obtain the solutions for \(x_1\), \(x_2\), and \(x_3\). Let me know if you'd like me to compute this further using Gaussian elimination in more detail.

---

Would you like detailed steps for solving the matrix, or is the explanation sufficient?

---

### Follow-up Questions:
1. What happens to the system if both equations are multiplied by the same constant?
2. How can we graphically interpret the different number of solutions for a system of equations?
3. What is the difference between consistent and inconsistent systems?
4. How can we extend these methods to larger systems of equations?
5. How does Gaussian elimination work for systems that lead to infinitely many solutions?

### Tip:
When solving systems of linear equations, checking if the equations are dependent or independent early on can save a lot of time!

2. Untuk nilai-nilai konstanta k manakah sistem persamaan linear berikut tidak mempunyai pemecahan? Persis satu pemecahan? Tak terhingga banyaknya pemecahan?
3. Pemecahan sistem linear di bawah ini adalah:

This problem involves determining when a system of linear equations has no solution, exactly one solution, or infinitely many solutions. The solution applies concepts of linear algebra, including Gaussian elimination and matrix methods. Suitable for students in Grades 10-12.

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