To solve the equation \( 3x + 4y - 6z = -60 \), we need more information, as it's currently a single equation with three unknowns. In general, to find the exact values of \( x \), \( y \), and \( z \), we would need two additional independent equations to form a system.

Without that, we can express one variable in terms of the others. For example, we can solve for \( z \) in terms of \( x \) and \( y \):

\[
-6z = -60 - 3x - 4y
\]

\[
z = \frac{60 + 3x + 4y}{6}
\]

This expresses \( z \) in terms of \( x \) and \( y \). We could do similar rearrangements for \( x \) or \( y \) if needed.

Would you like to provide more equations, or continue exploring this solution?

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Here are five related questions for further practice:

1. How would you solve for \( y \) in terms of \( x \) and \( z \)?
2. What happens if \( z = 0 \) in the original equation?
3. Can you find integer solutions for \( x \), \( y \), and \( z \) using this equation?
4. How would you graph this equation in three-dimensional space?
5. How does adding another equation change the possible solutions?

**Tip:** For systems of equations with multiple variables, using substitution or elimination methods can help find exact solutions!

Explore how to solve the linear equation 3x + 4y - 6z = -60 by expressing one variable in terms of the others. Understand algebraic techniques used in solving systems of equations and how degrees of freedom affect solutions. Ideal for students in Grades 9-11.

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