https://file.aiquickdraw.com/mathbot/file/17263279117868vtheftm.jpg

The equation from the image is:

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

Let's solve this equation step by step.

### Step 1: Find a common denominator

The denominators are \(x\) and \(x-6\), so the least common denominator (LCD) is \(x(x - 6)\). Rewrite both fractions with this common denominator:

\[
\frac{1}{x} = \frac{x-6}{x(x-6)}
\]
\[
\frac{1}{x-6} = \frac{x}{x(x-6)}
\]

So the equation becomes:

\[
\frac{x-6}{x(x-6)} + \frac{x}{x(x-6)} = \frac{1}{3}
\]

### Step 2: Combine the fractions on the left-hand side

Now, combine the fractions on the left-hand side:

\[
\frac{(x-6) + x}{x(x-6)} = \frac{1}{3}
\]

Simplify the numerator:

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

### Step 3: Cross-multiply

Now, cross-multiply to solve for \(x\):

\[
3(2x - 6) = x(x - 6)
\]

### Step 4: Expand both sides

Expand both sides of the equation:

\[
6x - 18 = x^2 - 6x
\]

### Step 5: Rearrange the equation

Move all terms to one side to set the equation to zero:

\[
x^2 - 12x + 18 = 0
\]

### Step 6: Solve the quadratic equation

Now, solve the quadratic equation using the quadratic formula:

\[
x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(18)}}{2(1)}
\]

Simplifying:

\[
x = \frac{12 \pm \sqrt{144 - 72}}{2}
\]
\[
x = \frac{12 \pm \sqrt{72}}{2}
\]
\[
x = \frac{12 \pm 6\sqrt{2}}{2}
\]
\[
x = 6 \pm 3\sqrt{2}
\]

### Final Answer

The solutions are:

\[
x = 6 + 3\sqrt{2} \quad \text{or} \quad x = 6 - 3\sqrt{2}
\]

Would you like any further explanation or details on any part of the solution?

### Related Questions:
1. How do you find the least common denominator between two rational expressions?
2. How does cross-multiplying help solve rational equations?
3. What are the steps for solving a quadratic equation?
4. Can all quadratic equations be solved using the quadratic formula?
5. What is the significance of the discriminant in the quadratic formula?

### Tip:
Always check your solutions to ensure they don't create any zero denominators in the original equation!

This page provides a step-by-step solution to the equation 1/x + 1/(x - 6) = 1/3. Learn how to simplify rational expressions, combine fractions, and solve the resulting quadratic equation using the quadratic formula. Suitable for Grades 9-12.

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