Let’s analyze and solve the equations step by step based on the uploaded image. Each equation requires finding the correct value of $z$ from the options provided (A to F).

1. Equation 1:

$\frac{2}{z - 3} = 1$

• Multiply through by $z - 3$ (as long as $z \neq 3$): $2 = z - 3$
• Solve for $z$: $z = 2 + 3 = 5$

Answer for 1: $z = 5$ (Option E).

2. Equation 2:

$\frac{5}{z - 1} = 2$

• Multiply through by $z - 1$ (as long as $z \neq 1$): $5 = 2(z - 1)$
• Expand and solve for $z$: $5 = 2z - 2 \quad \Rightarrow \quad 2z = 7 \quad \Rightarrow \quad z = \frac{7}{2} = 3.5$

$z = 3.5$ is not among the provided options, so it seems this specific equation might have a mismatch.

3. Equation 3:

$\frac{3}{z + 4} = \frac{1}{3}$

• Cross-multiply: $3 \cdot 3 = z + 4$
• Solve for $z$: $9 = z + 4 \quad \Rightarrow \quad z = 9 - 4 = 5$

Answer for 3: $z = 5$ (Option E).

4. Equation 4:

$\frac{5}{z - 6} = 2 \frac{1}{2} = \frac{5}{2}$

• Cross-multiply: $5 \cdot 2 = 5(z - 6)$
• Simplify and solve for $z$: $10 = 5z - 30 \quad \Rightarrow \quad 5z = 40 \quad \Rightarrow \quad z = \frac{40}{5} = 8$

$z = 8$ is not listed as a valid option, so further verification might be needed.

Final Answers:

1. $z = 5$ (Option E)
2. $z = 3.5$ (not listed)
3. $z = 5$ (Option E)
4. $z = 8$ (not listed)

Would you like clarification, or should I verify any equation further? Let me know!

5 Related Questions:

1. How do you solve fractional equations in general?
2. Why is it important to check for restrictions on $z$ (e.g., $z - 3 \neq 0$)?
3. Can we rewrite mixed numbers like $2 \frac{1}{2}$ directly as improper fractions?
4. What is the purpose of cross-multiplication in solving rational equations?
5. How would you handle equations with no solutions or extraneous roots?

Tip: Always verify your solution by substituting the value of $z$ back into the original equation!