Let's solve the equation:

\[
\frac{1}{3}(5t - 5) = \frac{11}{3}t - t - \frac{5}{6}
\]

### Step 1: Distribute the \(\frac{1}{3}\) on the left side.
\[
\frac{1}{3}(5t - 5) = \frac{1}{3} \times 5t - \frac{1}{3} \times 5 = \frac{5}{3}t - \frac{5}{3}
\]
So, the equation becomes:
\[
\frac{5}{3}t - \frac{5}{3} = \frac{11}{3}t - t - \frac{5}{6}
\]

### Step 2: Combine like terms on the right side.
We can rewrite the right-hand side:
\[
\frac{11}{3}t - t = \frac{11}{3}t - \frac{3}{3}t = \frac{8}{3}t
\]
Now the equation is:
\[
\frac{5}{3}t - \frac{5}{3} = \frac{8}{3}t - \frac{5}{6}
\]

### Step 3: Eliminate the fractions by multiplying through by 6 (the least common denominator of 3 and 6).
Multiplying every term by 6:
\[
6 \times \left( \frac{5}{3}t - \frac{5}{3} \right) = 6 \times \left( \frac{8}{3}t - \frac{5}{6} \right)
\]
This gives:
\[
2 \times (5t - 5) = 2 \times 8t - 5
\]
Simplifying:
\[
10t - 10 = 16t - 5
\]

### Step 4: Solve for \(t\).
Move all the \(t\)-terms to one side and the constants to the other side:
\[
10t - 16t = -5 + 10
\]
\[
-6t = 5
\]
Now, divide both sides by \(-6\):
\[
t = -\frac{5}{6}
\]

So, the solution is:
\[
t = -\frac{5}{6}
\]

Would you like any further details or explanations on any part of this solution?

### Related Questions:
1. How do you solve equations involving fractions and variables?
2. What are the properties of equality used in solving linear equations?
3. Can you explain how to distribute fractions across terms in parentheses?
4. How do you find the least common denominator when solving equations with fractions?
5. What are some common mistakes when solving multi-step equations?

**Tip:** Always aim to eliminate fractions early in an equation by multiplying by the least common denominator—it simplifies the steps!

Learn how to solve the linear equation 1/3(5t - 5) = 11/3t - t - 5/6 with fractions and variables. This step-by-step solution demonstrates the use of algebraic methods such as the distributive property and elimination of fractions, suitable for students in Grades 8-10.

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