find the solution of following simultainous equation x-y+z=2, -x+2+1, x+y-z=0

The system of simultaneous equations given is:

1. \( x - y + z = 2 \)
2. \( -x + 2 + 1 = 0 \)
3. \( x + y - z = 0 \)

Let's solve this step-by-step.

### Step 1: Simplify the equations

Simplify the second equation:
\[ -x + 2 + 1 = 0 \]
\[ -x + 3 = 0 \]
\[ x = 3 \]

Now we substitute \( x = 3 \) into the other two equations:

### Step 2: Substitute \( x = 3 \) into the first and third equations

1. \( 3 - y + z = 2 \)
2. \( 3 + y - z = 0 \)

### Step 3: Solve the new system of equations

Now we have the system:
\[ 3 - y + z = 2 \quad \text{(Equation 1)} \]
\[ 3 + y - z = 0 \quad \text{(Equation 2)} \]

Let's simplify these equations:

From Equation 1:
\[ 3 - y + z = 2 \]
\[ -y + z = -1 \]
\[ y - z = 1 \quad \text{(Equation 3)} \]

From Equation 2:
\[ 3 + y - z = 0 \]
\[ y - z = -3 \quad \text{(Equation 4)} \]

### Step 4: Compare Equation 3 and Equation 4

We have:
\[ y - z = 1 \]
\[ y - z = -3 \]

This is a contradiction, meaning the system of equations is inconsistent and has no solution.

### Conclusion

The given system of equations has no solution because the two derived equations \( y - z = 1 \) and \( y - z = -3 \) are contradictory.

Would you like more details or have any questions about the solution steps?

### Related questions to explore:
1. What are simultaneous equations?
2. How do you determine if a system of equations is inconsistent?
3. Can you provide another example of solving simultaneous equations?
4. What methods can be used to solve systems of equations?
5. What is the substitution method in solving systems of equations?
6. How does the elimination method work in solving systems of equations?
7. What is the graphical interpretation of solving simultaneous equations?
8. How can matrices be used to solve systems of equations?

#### Tip:
Always check each equation carefully when solving systems of equations to ensure consistency and avoid contradictions.

Learn how to solve the system of simultaneous equations x - y + z = 2, -x + 2 + 1 = 0, and x + y - z = 0 with a detailed step-by-step solution. Understand the process of simplifying, substituting, and identifying inconsistencies in simultaneous equations. Suitable for students in Grades 10-12.

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