In each of these questions, two equations I. and II. are give. you have to solve both the equations and give answer
I. ` (1)/(3) + (5)/( x^(2)) = (8)/( 3 x)`
II. ` ( y)/( 2) + (21)/( 2y) = 5`

Options

A. ` x gt y `
B. ` x lt y `
C. ` x ge y `
D. x = y or no relation

Answer

To solve the given equations step by step, we will start with Equation I and then proceed to Equation II.

### Equation I:
\[ \frac{1}{3} + \frac{5}{x^2} = \frac{8}{3x} \]

**Step 1:** Eliminate the fractions by cross-multiplying.
\[ 3x \left( \frac{1}{3} + \frac{5}{x^2} \right) = 8 \]
This simplifies to:
\[ x + \frac{15}{x} = 8 \]

**Step 2:** Multiply through by \(x\) to eliminate the fraction.
\[ x^2 + 15 = 8x \]

**Step 3:** Rearrange the equation into standard quadratic form.
\[ x^2 - 8x + 15 = 0 \]

**Step 4:** Factor the quadratic equation.
\[ (x - 3)(x - 5) = 0 \]

**Step 5:** Set each factor to zero and solve for \(x\).
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]

### Equation II:
\[ \frac{y}{2} + \frac{21}{2y} = 5 \]

**Step 1:** Eliminate the fractions by multiplying through by \(2y\).
\[ y^2 + 21 = 10y \]

**Step 2:** Rearrange the equation into standard quadratic form.
\[ y^2 - 10y + 21 = 0 \]

**Step 3:** Factor the quadratic equation.
\[ (y - 3)(y - 7) = 0 \]

**Step 4:** Set each factor to zero and solve for \(y\).
\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \]
\[ y - 7 = 0 \quad \Rightarrow \quad y = 7 \]

### Summary of Solutions:
From Equation I, we found:
- \(x = 3\) or \(x = 5\)

From Equation II, we found:
- \(y = 3\) or \(y = 7\)

### Relation Between \(x\) and \(y\):
- When \(x = 3\), \(y = 3\) (they are equal).
- When \(x = 5\), \(y = 3\) (here \(x > y\)).
- When \(x = 5\), \(y = 7\) (here \(y > x\)).

Thus, the relationship cannot be definitively established as \(x\) can be either smaller than, equal to, or greater than \(y\).

### Conclusion:
The answer is that there is no definitive relation between \(x\) and \(y\).

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