Let's solve the equation step by step:

Given equation:

\[
2z - 5 - 5z = 7 + 3z - 3
\]

### Step 1: Simplify both sides of the equation
First, combine like terms on each side:

**Left side:**

\[
2z - 5z - 5 = -3z - 5
\]

**Right side:**

\[
7 - 3 = 4 + 3z
\]

The equation now becomes:

\[
-3z - 5 = 4 + 3z
\]

### Step 2: Move all the \( z \)-terms to one side
Add \( 3z \) to both sides to eliminate the \( z \)-term from the left side:

\[
-5 = 4 + 6z
\]

### Step 3: Move the constant term to the other side
Subtract 4 from both sides:

\[
-5 - 4 = 6z
\]

\[
-9 = 6z
\]

### Step 4: Solve for \( z \)
Now divide both sides by 6:

\[
z = \frac{-9}{6} = \frac{-3}{2}
\]

### Final Answer:
\[
z = -\frac{3}{2}
\]

Let me know if you need further explanation or have any questions!

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Here are 5 related questions you might want to explore:
1. How do you solve equations with fractions like \( \frac{2}{3}z - \frac{5}{6} = 4 \)?
2. What happens when both sides of the equation have variable terms with different powers?
3. How do you handle equations with parentheses, like \( 2(z - 3) = 5 + 4z \)?
4. Can we have multiple solutions for a linear equation like this one?
5. What is the geometric interpretation of a solution to a linear equation in terms of graphing?

**Tip:** Always combine like terms first to simplify the equation and reduce calculation complexity.

This guide walks through solving the linear equation 2z - 5 - 5z = 7 + 3z - 3 step-by-step, including combining like terms and isolating variables. Perfect for students in Grades 8-10 who are learning algebraic concepts like linear equations and simplification.

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