Given the equation 1/x = 2/3x - 1, solve for the value of x.

To solve the equation \( \frac{1}{x} = \frac{2}{3x} - 1 \) for \( x \), follow these steps:

### Step 1: Eliminate the fractions by multiplying the entire equation by \( 3x \) (the least common denominator):
\[
3x \cdot \frac{1}{x} = 3x \cdot \left(\frac{2}{3x} - 1\right)
\]
This simplifies to:
\[
3 = 2 - 3x
\]

### Step 2: Isolate the \( x \) term by subtracting 2 from both sides:
\[
3 - 2 = -3x
\]
\[
1 = -3x
\]

### Step 3: Solve for \( x \) by dividing both sides by -3:
\[
x = -\frac{1}{3}
\]

### Solution:
The value of \( x \) is \( x = -\frac{1}{3} \).

Would you like further details on this solution or have any questions?

Here are 5 related questions you might explore:
1. What happens if we substitute \( x = -\frac{1}{3} \) back into the original equation?
2. How would the solution change if the equation was \( \frac{1}{x} = \frac{2}{3x} + 1 \)?
3. What is the graphical interpretation of this equation?
4. Can you solve a similar equation where the variables are in the denominators on both sides?
5. How would the steps change if you encounter an equation of the form \( \frac{a}{x} = \frac{b}{cx} - d \)?

**Tip:** Always check your solution by substituting it back into the original equation to verify its correctness.

Learn how to solve the equation 1/x = 2/3x - 1 step-by-step with a detailed solution. This problem covers key algebraic concepts, including equations involving fractions and algebraic manipulation. Suitable for students in Grades 9-12.

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