Understanding the Equations
The given equations are:
- Xy / (x + y) = 20
- Yz / (y + z) = 40
- Zx / (z + x) = 24
These equations relate three variables X, Y, and Z. Each equation expresses a ratio involving products of two variables divided by their sum, resulting in a constant.
Analyzing Each Equation
- First Equation: Xy / (x + y) = 20
- This implies that the product of X and Y is balanced by their sum to yield 20.
- Rearranging gives us: Xy = 20(x + y).
- Second Equation: Yz / (y + z) = 40
- Similar to the first, this means Y and Z together yield a value of 40 when considering their ratio.
- Rearranging yields: Yz = 40(y + z).
- Third Equation: Zx / (z + x) = 24
- This indicates a relationship between Z and X, producing a value of 24.
- Rearranging gives us: Zx = 24(z + x).
Finding Relationships
To solve these equations, we can express each variable in terms of the others or find a common value for X, Y, and Z.
Using the rearranged forms, we can derive:
- X = (20(x + y))/y
- Y = (40(y + z))/z
- Z = (24(z + x))/x
By substituting these expressions into one another, we can find specific values for each variable.
Conclusion
These equations can be solved systematically to find the values of X, Y, and Z. Understanding the relationships in these types of problems is crucial for solving similar equations in competitive exams like UPSC.