To solve the system of simultaneous equations given:

1. $x - y + z = 2$
2. $-x + 2 + 1 = 0$
3. $x + y - z = 0$

We need to follow these steps:

Step 1: Simplify the equations

Let's simplify the second equation first: $-x + 2 + 1 = 0$ $-x + 3 = 0$ $x = 3$

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: $3 - y + z = 2 \quad \text{(Equation 1)}$ $3 + y - z = 0 \quad \text{(Equation 2)}$

Subtract Equation 1 from Equation 2: $(3 + y - z) - (3 - y + z) = 0 - 2$ $3 + y - z - 3 + y - z = -2$ $2y - 2z = -2$ $y - z = -1 \quad \text{(Equation 3)}$

Step 4: Substitute Equation 3 into Equation 1

$3 - y + z = 2$ From Equation 3, we know: $y = z - 1$

Substitute $y = z - 1$ into Equation 1: $3 - (z - 1) + z = 2$ $3 - z + 1 + z = 2$ $4 = 2$ This indicates a contradiction, meaning there might have been an error in the original problem statement or simplification.

Revisiting the equations to find a consistent solution, let's verify:

Recheck the simplified equations: $x = 3$ $3 - y + z = 2 \rightarrow y = z - 1$

Hence, we can solve these correctly:

1. $3 - y + z = 2$
2. $y = z - 1$

$y = z - 1$

Substitute back: $3 - (z - 1) + z = 2$ $3 - z + 1 + z = 2$ $4 = 2$

These steps confirm the contradiction.

Hence no solution exists as they represent inconsistent equations.

Confirmation of no solution:

These equations have no common solution as rechecking proves inconsistent result.

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 double-check each step when solving systems of equations to ensure consistency and accuracy.