https://file.aiquickdraw.com/mathbot/file/1725581928953d7b9ov5d.jpg

The system of equations shown in the image is:

\[
\begin{aligned}
x + y + z &= 3 \\
x + 2y - z &= 2
\end{aligned}
\]

You are asked to solve this system of equations, determining the values of \( x \), \( y \), and \( z \). If there are infinitely many solutions, you must express your solution in terms of one variable (for example, \( z = t \)).

I'll solve this step by step.

1. **Subtract the second equation from the first** to eliminate one variable, \( x \):
   \[
   (x + y + z) - (x + 2y - z) = 3 - 2
   \]
   This simplifies to:
   \[
   y + z + z = 1 \implies y + 2z = 1
   \]
   So, the third equation we have is:
   \[
   y + 2z = 1
   \]

2. Now we have:
   \[
   y + 2z = 1 \quad \text{(Equation 3)}
   \]
   \[
   x + y + z = 3 \quad \text{(Equation 1)}
   \]

3. **Solve for \( y \) in terms of \( z \)** from Equation 3:
   \[
   y = 1 - 2z
   \]

4. **Substitute \( y = 1 - 2z \) into Equation 1** to find \( x \) in terms of \( z \):
   \[
   x + (1 - 2z) + z = 3
   \]
   Simplifying:
   \[
   x + 1 - z = 3 \implies x = 2 + z
   \]

5. Therefore, the solution for this system in terms of \( z \) (since there are infinitely many solutions) is:
   \[
   x = 2 + z, \quad y = 1 - 2z, \quad z = z
   \]
   where \( z \) is a free parameter.

Would you like further details or clarifications?

Here are five questions related to this problem:

1. What does it mean when a system has infinitely many solutions?
2. Can this system of equations be solved using matrices? How?
3. How do we check if a system of equations is dependent or independent?
4. How do we express solutions in parametric form?
5. What are other methods to solve systems of linear equations?

**Tip:** Always verify your solution by substituting it back into the original equations to ensure correctness.

Find all solutions of the given system of equations and write your answer explicitly. If there is no solution, state 'NO SOLUTION'. If the system is dependent, express your answer in terms of a free variable, t (or z).

This solution covers solving a system of linear equations with infinitely many solutions, expressed in terms of a free variable. Learn the step-by-step process, including subtracting equations, solving for variables, and expressing the solution parametrically. Suitable for students in Grades 9-12.

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