What is equal to?
Simplifying Algebraic Expression with x, y, z
The problem asks us to simplify the following complex algebraic expression:
\[ \frac{\frac{x}{x-y}+\frac{y}{y-z}+\frac{z}{z-x}} {\frac{x+y}{x-y}+\frac{y+z}{y-z}+\frac{z+x}{z-x}+3} \]Let's denote the numerator as \( N \) and the denominator as \( D \).
Numerator \( N \):
\[ N = \frac{x}{x-y}+\frac{y}{y-z}+\frac{z}{z-x} \]Denominator \( D \):
\[ D = \frac{x+y}{x-y}+\frac{y+z}{y-z}+\frac{z+x}{z-x}+3 \]Our goal is to simplify the fraction \( \frac{N}{D} \).
Analyzing the Denominator Structure
Let's examine the terms in the denominator \( D \). We can rewrite the denominator by adding 1 to each of the first three terms:
- Consider the first term and add 1: \[ \frac{x+y}{x-y} + 1 = \frac{x+y + (x-y)}{x-y} = \frac{x+y+x-y}{x-y} = \frac{2x}{x-y} \]
- Consider the second term and add 1: \[ \frac{y+z}{y-z} + 1 = \frac{y+z + (y-z)}{y-z} = \frac{y+z+y-z}{y-z} = \frac{2y}{y-z} \]
- Consider the third term and add 1: \[ \frac{z+x}{z-x} + 1 = \frac{z+x + (z-x)}{z-x} = \frac{z+x+z-x}{z-x} = \frac{2z}{z-x} \]
Now, let's rewrite the denominator \( D \) using these results. The original denominator is:
\[ D = \left(\frac{x+y}{x-y}\right) + \left(\frac{y+z}{y-z}\right) + \left(\frac{z+x}{z-x}\right) + 3 \]We can group the terms like this:
\[ D = \left(\frac{x+y}{x-y} + 1\right) + \left(\frac{y+z}{y-z} + 1\right) + \left(\frac{z+x}{z-x} + 1\right) \]Substituting the results from above:
\[ D = \left(\frac{2x}{x-y}\right) + \left(\frac{2y}{y-z}\right) + \left(\frac{2z}{z-x}\right) \]We can factor out a 2:
\[ D = 2 \left( \frac{x}{x-y} + \frac{y}{y-z} + \frac{z}{z-x} \right) \]Final Simplification
We recognize that the expression inside the parentheses is exactly the numerator \( N \).
\[ D = 2 \times N \]Now we can substitute this back into the original fraction \( \frac{N}{D} \):
\[ \frac{N}{D} = \frac{N}{2N} \]Assuming \( N \neq 0 \), we can cancel \( N \) from the numerator and the denominator:
\[ \frac{N}{2N} = \frac{1}{2} \]Therefore, the simplified value of the given algebraic expression is \( \frac{1}{2} \).
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