Determine whether (3, -3, -5) is the solution to the system of equations.  State why or why not.
3x + 5y + 2z = 0
x + y = 0
6x - 7y + z = 1


To determine whether the point \((3, -3, -5)\) is a solution to the given system of equations, we will substitute \(x = 3\), \(y = -3\), and \(z = -5\) into each equation and check if the resulting expressions satisfy the equations.

The system of equations is:

1. \(3x + 5y + 2z = 0\)
2. \(x + y = 0\)
3. \(6x - 7y + z = 1\)

### Substituting \((3, -3, -5)\) into each equation:

#### Equation 1: \(3x + 5y + 2z = 0\)

Substitute \(x = 3\), \(y = -3\), \(z = -5\):

\[
3(3) + 5(-3) + 2(-5) = 0
\]

Calculate:

\[
9 - 15 - 10 = 0
\]

\[
-16 \neq 0 \quad \text{(False)}
\]

Since this equation is not satisfied, \((3, -3, -5)\) is not a solution to the system of equations. However, let's check the other equations to confirm consistency:

#### Equation 2: \(x + y = 0\)

Substitute \(x = 3\), \(y = -3\):

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

\[
0 = 0 \quad \text{(True)}
\]

#### Equation 3: \(6x - 7y + z = 1\)

Substitute \(x = 3\), \(y = -3\), \(z = -5\):

\[
6(3) - 7(-3) + (-5) = 1
\]

Calculate:

\[
18 + 21 - 5 = 1
\]

\[
34 \neq 1 \quad \text{(False)}
\]

### Conclusion

Since the point \((3, -3, -5)\) does not satisfy the first and third equations (they do not yield true statements), \((3, -3, -5)\) **is not a solution** to the system of equations.

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

### Related Questions:
1. How can we determine if a system of equations has no solution, one solution, or infinitely many solutions?
2. What are the methods to solve a system of linear equations with three variables?
3. What is the significance of a solution in terms of intersection points of planes in 3D geometry?
4. How can graphing help in visualizing solutions to systems of equations?
5. How do the coefficients in a system of equations influence its solutions?

**Tip:** Always check every equation in the system to ensure a proposed solution satisfies all of them. If any equation does not hold, the proposed point is not a solution.

Explore whether (3, -3, -5) satisfies a system of linear equations involving three variables. Check each equation for consistency and learn how to verify solutions. Suitable for high school students studying linear algebra.

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