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How to solve this equation and get infinite solutions

The system of linear equations in the image is as follows:

\[
1. \ x + y + z = 2
\]
\[
2. \ x + y - z = 2
\]
\[
3. \ 2x + 2y + z = 4
\]

To determine whether this system has infinite solutions, we can analyze it using methods such as substitution, elimination, or matrix representation. Let's start by solving this system step by step.

### Step 1: Subtract Equation 2 from Equation 1
Subtracting the second equation from the first:

\[
(x + y + z) - (x + y - z) = 2 - 2
\]

Simplifying this:

\[
2z = 0 \quad \Rightarrow \quad z = 0
\]

### Step 2: Substitute \(z = 0\) into the other equations
Now that we know \(z = 0\), substitute this value into the remaining two equations.

From Equation 1:
\[
x + y + 0 = 2 \quad \Rightarrow \quad x + y = 2
\]

From Equation 3:
\[
2x + 2y + 0 = 4 \quad \Rightarrow \quad 2x + 2y = 4
\]

### Step 3: Simplify Equation 3
The third equation simplifies to:
\[
x + y = 2
\]

This is the same as the equation we obtained from the first equation after substituting \(z = 0\). 

### Conclusion: Infinite Solutions
Since both equations \(x + y = 2\) are identical, and we have no further restrictions, \(x\) and \(y\) can vary as long as they satisfy \(x + y = 2\). This means there are infinitely many solutions where \(z = 0\) and \(x + y = 2\).

### General Solution:
- \(z = 0\)
- \(x = t\) and \(y = 2 - t\), where \(t\) is any real number.

Therefore, this system of equations has **infinite solutions**.

Would you like more details, or do you have any questions about this method?

### Related Questions:
1. How can we represent the solution set of this system graphically?
2. What conditions must a system of linear equations meet to have infinite solutions?
3. Can we determine a specific solution if additional constraints are added to this system?
4. How does the rank of the coefficient matrix relate to the number of solutions?
5. What is the difference between consistent and inconsistent systems of linear equations?

### Tip:
When dealing with systems of equations, always check if the equations are dependent (i.e., multiples of each other) as this often leads to infinite solutions.

Explore how to solve a system of linear equations with infinite solutions step-by-step. Understand the concept of infinite solutions in linear systems and learn practical methods such as substitution and elimination. Suitable for advanced high school students and anyone interested in linear algebra.

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