Solving Mathematical Puzzles: x3 - y3 120 and x2y2 24 - xy
Introduction to Algebraic Equations
Algebraic equations are fundamental building blocks for solving complex mathematical puzzles and real-world problems. In this article, we delve into a specific problem that will require us to apply various algebraic techniques to derive a solution. The problem at hand involves a pair of equations: x3 - y3 120 and x2y2 24 - xy. Let's explore these step by step.
Step 1: Simplifying the First Equation
The first equation given is x3 - y3 120. This is a difference of cubes, which can be expanded using the identity:
x3 - y3 (x - y)(x2 xy y2)
Therefore, we can rewrite the equation as:
(x - y)(x2 xy y2) 120
Step 2: Simplifying the Second Equation
The second equation is x2y2 24 - xy. We can rearrange it as:
x2y2 xy 24
Notice that this can be written as:
xy(x1y1 1) 24
Since x1y1 xy, we can simplify further to:
xy(x y) 24
Step 3: Combining and Simplifying
Now we combine the two simplified equations. Recall from the first equation:
(x - y)(x2 xy y2) 120
We also know from the second equation:
xy(x y) 24
Let's assume x - y k, then:
(x - y)(x2 xy y2) 120
This simplifies to:
k(x2 xy y2) 120
Since xy(x y) 24, we can substitute x y as a factor in x2 xy y2:
xy(x y) 24
Divide through by xy:
x y 24/xy
Now, substituting this back into x2 xy y2 gives:
x2 xy y2 (x y)2 - xy
Thus, we get:
(x - y)((x y)2 - xy) 120
We know from xy(x y) 24 that:
(x - y)(24/xy - xy) 120
Let's simplify further:
24(x - y)/xy - xy(x - y) 120
Dividing through by x - y
24/xy - xy 120/(x - y)
Since x - y 5 from our problem statement, we get:
24/xy - xy 24/5
Let's solve for xy:
24 - (5xy) 24
Thus, we can deduce:
x - y 5
Conclusion
By applying algebraic techniques, we have successfully solved the given puzzle. The final answer is:
x - y 5
This example demonstrates the power of algebraic manipulation and substitution in solving complex equations. For more such problems and techniques in algebra, keep exploring mathematical puzzles and challenges.