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To solve for \( z \) in the equation:

\[
-cz + 6z = tz + 83
\]

we'll start by combining like terms and isolating \( z \).

### Step 1: Combine Terms with \( z \)

Rewrite the equation as:
\[
(-c + 6)z = tz + 83
\]

Move \( tz \) to the left side to bring all \( z \)-terms together:
\[
(-c + 6 - t)z = 83
\]

### Step 2: Solve for \( z \)

Assuming \( -c + 6 - t \neq 0 \), divide both sides by \( -c + 6 - t \) to isolate \( z \):
\[
z = \frac{83}{-c + 6 - t}
\]

### Final Answer

\[
z = \frac{83}{6 - c - t}
\]

Let me know if you want further details or have any questions!

Here are some related questions that might expand your understanding:
1. How do we handle cases where \( -c + 6 - t = 0 \)?
2. Can this method be applied if the equation had more terms?
3. What happens if \( c \) or \( t \) is negative or zero?
4. How would this equation change if \( 83 \) were a variable instead of a constant?
5. Could we use substitution methods for similar equations with multiple variables?

**Tip:** When solving for a variable, always bring like terms together first to simplify the equation.

Solve for \( z \): \(-cz + 6z = tz + 83\)

This page provides a step-by-step solution to solve for z in the linear equation -cz + 6z = tz + 83. Key algebraic techniques, such as combining like terms and isolating variables, are used in the solution, suitable for high school students in Grades 10-12.

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