Determine whether (0, 1, -1) is the solution to the system of equations.  State why or why not.
2x − 6y + 6z = −12
x + 4y + 5z = −1
−x + 2y + 3z = −1


To determine whether the point \((0, 1, -1)\) is a solution to the given system of equations, we need to substitute \(x = 0\), \(y = 1\), and \(z = -1\) into each equation and check if the equations hold true.

The system of equations is:

1. \(2x - 6y + 6z = -12\)
2. \(x + 4y + 5z = -1\)
3. \(-x + 2y + 3z = -1\)

### Substituting \((0, 1, -1)\) into each equation:

#### Equation 1: \(2x - 6y + 6z = -12\)

Substitute \(x = 0\), \(y = 1\), \(z = -1\):

\[
2(0) - 6(1) + 6(-1) = -12
\]

Calculate:

\[
0 - 6 - 6 = -12
\]

\[
-12 = -12 \quad \text{(True)}
\]

#### Equation 2: \(x + 4y + 5z = -1\)

Substitute \(x = 0\), \(y = 1\), \(z = -1\):

\[
0 + 4(1) + 5(-1) = -1
\]

Calculate:

\[
0 + 4 - 5 = -1
\]

\[
-1 = -1 \quad \text{(True)}
\]

#### Equation 3: \(-x + 2y + 3z = -1\)

Substitute \(x = 0\), \(y = 1\), \(z = -1\):

\[
-(0) + 2(1) + 3(-1) = -1
\]

Calculate:

\[
0 + 2 - 3 = -1
\]

\[
-1 = -1 \quad \text{(True)}
\]

### Conclusion

Since substituting \((0, 1, -1)\) into each equation results in a true statement for all three equations, \((0, 1, -1)\) **is indeed a solution** to the system of equations.

Would you like to explore further details or have any questions?

### Related Questions:
1. What are the steps to solve a system of linear equations using substitution?
2. How do you solve a system of linear equations using the elimination method?
3. What is the geometric interpretation of a solution to a system of three equations in three variables?
4. Can a system of linear equations have more than one solution? If so, under what conditions?
5. How can we use matrices to solve a system of linear equations?

**Tip:** When verifying a solution to a system of equations, always substitute the values back into the original equations to ensure they satisfy all the equations.

Learn how to verify whether (0, 1, -1) is a solution to the system of linear equations 2x - 6y + 6z = -12, x + 4y + 5z = -1, and -x + 2y + 3z = -1. Detailed step-by-step verification process provided.

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