Make $$ g $$ the subject of $$ e=\sqrt{\frac{7 g+5}{11+2 g}} $$

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UpStudy AI · Accepted Answer

To make $$ g $$ the subject of the equation $$ e = \sqrt{\frac{7g + 5}{11 + 2g}} $$, follow these steps:

1. **Square Both Sides**  
   Start by eliminating the square root by squaring both sides of the equation:
   $$
   e^2 = \frac{7g + 5}{11 + 2g}
   $$

2. **Multiply Both Sides by the Denominator**  
   This removes the fraction:
   $$
   e^2 (11 + 2g) = 7g + 5
   $$
   $$
   11e^2 + 2e^2g = 7g + 5
   $$

3. **Gather Like Terms**  
   Bring all terms containing $$ g $$ to one side and the constant terms to the other:
   $$
   2e^2g - 7g = 5 - 11e^2
   $$

4. **Factor Out $$ g $$**  
   Factor $$ g $$ from the left side:
   $$
   g(2e^2 - 7) = 5 - 11e^2
   $$

5. **Solve for $$ g $$**  
   Finally, isolate $$ g $$ by dividing both sides by $$ (2e^2 - 7) $$:
   $$
   g = \frac{5 - 11e^2}{2e^2 - 7}
   $$
   To present the equation with a positive denominator, you can multiply both the numerator and the denominator by $$-1$$:
   $$
   g = \frac{11e^2 - 5}{7 - 2e^2}
   $$

**Final Answer:**
$$
g = \frac{11\,e^{2} - 5}{\,7\; -\; 2\,e^{2}\,}
$$
$$ g = \frac{11e^{2} - 5}{7 - 2e^{2}} $$

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