To solve the equation:

\[
\frac{x-1}{6} - \frac{x-3}{2} = -1
\]

### Step 1: Eliminate the fractions
The least common denominator (LCD) of 6 and 2 is 6. Multiply every term by 6 to clear the denominators:

\[
6 \times \left( \frac{x-1}{6} \right) - 6 \times \left( \frac{x-3}{2} \right) = 6 \times (-1)
\]

This simplifies to:

\[
(x-1) - 3(x-3) = -6
\]

### Step 2: Expand and simplify the terms
Distribute the \(3\) in the second term:

\[
(x-1) - 3x + 9 = -6
\]

Simplify the left-hand side:

\[
x - 1 - 3x + 9 = -6
\]

Combine like terms:

\[
-2x + 8 = -6
\]

### Step 3: Solve for \(x\)
Isolate the \(x\)-term:

\[
-2x = -6 - 8
\]
\[
-2x = -14
\]

Now, divide by \(-2\):

\[
x = \frac{-14}{-2} = 7
\]

### Final Answer:
\[
x = 7
\]

Would you like further clarification or details on any step?

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Here are some related questions for further practice:
1. Solve \(\frac{x-2}{3} - \frac{x+1}{5} = 1\).
2. Solve \(\frac{2x+1}{4} - \frac{x-3}{2} = 2\).
3. Solve \( \frac{3x}{7} + \frac{x+2}{5} = 4 \).
4. Solve \(\frac{2x-5}{9} = \frac{4x+1}{3}\).
5. Solve \(\frac{4x+1}{2} - \frac{x-4}{3} = 0\).

**Tip:** Always check your solutions by substituting the value of \(x\) back into the original equation to ensure correctness.

Learn how to solve the linear equation (x-1)/6 - (x-3)/2 = -1 with a step-by-step guide. This problem covers essential algebraic concepts like clearing fractions, combining like terms, and isolating the variable. Perfect for students in Grades 8-10.

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