Solve the following linear system x + y + z = 1, 2x + 2y + 2z = 2 and 3x + 3y + 3z = 3.

Let's solve the given system of linear equations:

\[
\begin{aligned}
1. \quad & x + y + z &= 1 \quad \text{(Equation 1)} \\
2. \quad & 2x + 2y + 2z &= 2 \quad \text{(Equation 2)} \\
3. \quad & 3x + 3y + 3z &= 3 \quad \text{(Equation 3)}
\end{aligned}
\]

### Step 1: Simplify the system

Notice that the second and third equations are multiples of the first equation:

- Equation 2: \(2(x + y + z) = 2\), which simplifies to \(x + y + z = 1\).
- Equation 3: \(3(x + y + z) = 3\), which also simplifies to \(x + y + z = 1\).

This means all three equations are equivalent and represent the same plane in three-dimensional space.

### Step 2: Interpret the solution

Since all the equations are identical, the system is **dependent** and has infinitely many solutions. The solution can be expressed by choosing two of the variables as free parameters and solving for the third.

### Step 3: Parametrize the solution

Let's express the solution in parametric form by solving for \(x\) and \(y\) in terms of \(z\):

\[
x + y = 1 - z
\]

Let \(z = t\), where \(t\) is a free parameter. Then, we have:

\[
x + y = 1 - t
\]

Let's express \(x\) in terms of \(y\) and \(t\):

\[
x = 1 - t - y
\]

Thus, the general solution is:

\[
\begin{aligned}
x &= 1 - t - s, \\
y &= s, \\
z &= t,
\end{aligned}
\]

where \(s\) and \(t\) are free parameters.

### Final Answer:
The solution set of the system is \((x, y, z) = (1 - s - t, s, t)\), where \(s\) and \(t\) are real numbers. This represents an infinite number of solutions.

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Would you like further details on any specific part of this solution? Here are some related questions that might interest you:

1. How do you determine if a system of linear equations has no solution?
2. What is the significance of dependent equations in a system?
3. How can we apply matrix methods to solve linear systems?
4. What is the geometric interpretation of a system with infinite solutions?
5. How does the rank of a matrix relate to the number of solutions in a linear system?
6. How to solve a linear system using the substitution method?
7. What are the real-world applications of solving linear systems?
8. What is the difference between consistent and inconsistent systems?

**Tip:** When solving a system of linear equations, check if the equations are multiples of each other. This can simplify the system and help identify whether it has infinite solutions, no solutions, or a unique solution.

Learn how to solve the system of linear equations x + y + z = 1, 2x + 2y + 2z = 2, 3x + 3y + 3z = 3 with a step-by-step solution. Understand the concept of dependent systems and parametric solutions. Suitable for students in Grades 9-12 studying algebra and linear equations.

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