Question

solve for u.
\\(\frac{u - 1}{9u} + \frac{1}{6} = \frac{1}{u}\\)
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.

Explanation:

Step1: Find the common denominator

The denominators are \(9u\), \(6\), and \(u\). The least common denominator (LCD) of \(9u\), \(6\), and \(u\) is \(18u\). Multiply each term by \(18u\) to eliminate the denominators:

$$ 18u \cdot \frac{u - 1}{9u} + 18u \cdot \frac{1}{6} = 18u \cdot \frac{1}{u} $$

Step2: Simplify each term

Simplify each term after multiplying by \(18u\):

• For the first term: \(18u \cdot \frac{u - 1}{9u} = 2(u - 1)\)
• For the second term: \(18u \cdot \frac{1}{6} = 3u\)
• For the third term: \(18u \cdot \frac{1}{u} = 18\)

So the equation becomes:

$$ 2(u - 1) + 3u = 18 $$

Step3: Expand and combine like terms

Expand \(2(u - 1)\): \(2u - 2 + 3u = 18\)
Combine like terms: \(5u - 2 = 18\)

Step4: Solve for \(u\)

Add \(2\) to both sides: \(5u = 18 + 2 = 20\)
Divide both sides by \(5\): \(u = \frac{20}{5} = 4\)

Step5: Check for extraneous solutions

We need to check if \(u = 4\) makes the original denominators zero. The denominators are \(9u = 9\times4 = 36\), \(6\), and \(u = 4\), none of which are zero. So \(u = 4\) is a valid solution.

Answer:

\(4\)