Simplify the complex fraction. Assume no division by 0. + |N|X 2 4x y
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Simplify the complex problem assume no division by 0
![**Simplifying Complex Fractions**
**Problem:**
Simplify the complex fraction. Assume no division by 0.
\[
\frac{\frac{5}{x} + \frac{4x}{y}}{\frac{2}{x}}
\]
**Solution Steps:**
1. **Identify the Components:**
- The numerator of the complex fraction is \(\frac{5}{x} + \frac{4x}{y}\).
- The denominator of the complex fraction is \(\frac{2}{x}\).
2. **Simplify the Numerator:**
- To add \(\frac{5}{x}\) and \(\frac{4x}{y}\), find a common denominator: \(xy\).
- Rewrite the fractions:
\[
\frac{5y}{xy} + \frac{4x^2}{xy} = \frac{5y + 4x^2}{xy}
\]
3. **Form the New Complex Fraction:**
\[
\frac{\frac{5y + 4x^2}{xy}}{\frac{2}{x}}
\]
4. **Simplify the Complex Fraction:**
- Multiply by the reciprocal of the denominator:
\[
\frac{5y + 4x^2}{xy} \times \frac{x}{2} = \frac{(5y + 4x^2)x}{2xy}
\]
- Simplify by canceling \(x\) from the numerator and denominator:
\[
\frac{5y + 4x^2}{2y}
\]
5. **Final Simplified Form:**
\[
\frac{5y + 4x^2}{2y}
\]
This yields the simplified form of the original complex fraction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F28b27689-10ed-4307-b3bd-ec96e81b0de9%2F6cbb871a-5aad-4d64-ad81-3bb01e9877b5%2Fpbfjzmp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Simplifying Complex Fractions**
**Problem:**
Simplify the complex fraction. Assume no division by 0.
\[
\frac{\frac{5}{x} + \frac{4x}{y}}{\frac{2}{x}}
\]
**Solution Steps:**
1. **Identify the Components:**
- The numerator of the complex fraction is \(\frac{5}{x} + \frac{4x}{y}\).
- The denominator of the complex fraction is \(\frac{2}{x}\).
2. **Simplify the Numerator:**
- To add \(\frac{5}{x}\) and \(\frac{4x}{y}\), find a common denominator: \(xy\).
- Rewrite the fractions:
\[
\frac{5y}{xy} + \frac{4x^2}{xy} = \frac{5y + 4x^2}{xy}
\]
3. **Form the New Complex Fraction:**
\[
\frac{\frac{5y + 4x^2}{xy}}{\frac{2}{x}}
\]
4. **Simplify the Complex Fraction:**
- Multiply by the reciprocal of the denominator:
\[
\frac{5y + 4x^2}{xy} \times \frac{x}{2} = \frac{(5y + 4x^2)x}{2xy}
\]
- Simplify by canceling \(x\) from the numerator and denominator:
\[
\frac{5y + 4x^2}{2y}
\]
5. **Final Simplified Form:**
\[
\frac{5y + 4x^2}{2y}
\]
This yields the simplified form of the original complex fraction.
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