Let's solve the equation step by step:

Given equation:

\[
2Z - \frac{4}{3} + 3 - \frac{7}{5} = 2 - \frac{6}{6}
\]

### Step 1: Simplify the constants on both sides
First, simplify the fractions and the constants.

\[
\frac{6}{6} = 1, \quad \text{so the equation becomes:} 
\]
\[
2Z - \frac{4}{3} + 3 - \frac{7}{5} = 2 - 1
\]

Which simplifies to:

\[
2Z - \frac{4}{3} + 3 - \frac{7}{5} = 1
\]

### Step 2: Combine the constant terms on the left-hand side
We need to combine the constants \( 3 \), \( -\frac{4}{3} \), and \( -\frac{7}{5} \). To do this, let's first find a common denominator for the fractions.

- The least common denominator (LCD) for 3 and 5 is 15.

Rewrite the fractions:

\[
- \frac{4}{3} = -\frac{20}{15}, \quad -\frac{7}{5} = -\frac{21}{15}, \quad 3 = \frac{45}{15}
\]

Now combine them:

\[
\frac{45}{15} - \frac{20}{15} - \frac{21}{15} = \frac{45 - 20 - 21}{15} = \frac{4}{15}
\]

So the equation becomes:

\[
2Z + \frac{4}{15} = 1
\]

### Step 3: Solve for \( Z \)
Isolate \( 2Z \) by subtracting \( \frac{4}{15} \) from both sides:

\[
2Z = 1 - \frac{4}{15}
\]

Convert 1 to a fraction with denominator 15:

\[
1 = \frac{15}{15}
\]

Now subtract:

\[
2Z = \frac{15}{15} - \frac{4}{15} = \frac{11}{15}
\]

### Step 4: Divide both sides by 2
To solve for \( Z \):

\[
Z = \frac{\frac{11}{15}}{2} = \frac{11}{15} \times \frac{1}{2} = \frac{11}{30}
\]

### Final Answer:
\[
Z = \frac{11}{30}
\]

Let me know if you'd like further details on any of the steps!

Here are some related questions to consider:
1. How do you solve similar equations with different fractions?
2. What happens if the variable \( Z \) has higher powers?
3. How can you apply these steps to equations with decimals?
4. How does the process change if the fractions have negative denominators?
5. How do you handle equations where \( Z \) appears on both sides?

**Tip:** Always look for ways to simplify fractions early on to make solving equations easier!

This page provides a detailed, step-by-step solution to the linear equation 2Z - 4/3 + 3 - 7/5 = 2 - 6/6. Key algebraic concepts such as combining fractions and isolating variables are explained. Suitable for students in Grades 7-9.

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