Question

1. \\(\frac{1}{3r} + \frac{r - 2}{r} = \frac{2}{3r}\\)

Explanation:

Step1: Find common denominator

The denominators are \(3r\), \(r\), and \(3r\). The common denominator is \(3r\). Multiply each term by \(3r\) to eliminate denominators:
\(3r\times\frac{1}{3r}+3r\times\frac{r - 2}{r}=3r\times\frac{2}{3r}\)
Simplify each term: \(1 + 3(r - 2)=2\)

Step2: Expand and simplify

Expand \(3(r - 2)\): \(1+3r - 6 = 2\)
Combine like terms: \(3r - 5 = 2\)

Step3: Solve for \(r\)

Add 5 to both sides: \(3r=2 + 5\)
\(3r = 7\)
Divide by 3: \(r=\frac{7}{3}\)

Step4: Check for restrictions

Denominator \(r
eq0\) and \(3r
eq0\) (so \(r
eq0\)). \(\frac{7}{3}
eq0\), so it's valid.

Answer:

\(r = \frac{7}{3}\)