Introduction to Solving Real Number Equations

When dealing with equations involving real numbers, it's often necessary to methodically analyze and simplify the given conditions. In this article, we will explore how to solve a set of equations involving real numbers, using algebraic manipulations and properties of symmetric sums. The key to unlocking the solution lies in a systematic approach that utilizes the given conditions to derive the values of the variables.

Given Equations and Initial Steps

We start with the following system of equations:

begin{cases} abcd 20 abacadbcbdcd 150 end{cases}

To simplify this, we introduce new variables:

[x a - 5, ; y b - 5, ; z c - 5, ; t d - 5]

Substituting these into the original equations, we get:

[ (x 5)(y 5)(z 5)(t 5) 20 ]

[ (x 5)(y 5)(z 5)(t 5)(x 5)(y 5)(z 5)(t 5) 150 ]

Notice that the second equation is the first equation squared:

[ (x 5)(y 5)(z 5)(t 5) sqrt{150} ]

Since (sqrt{150} sqrt{20^2 - 2 times 150} 10), we have:

[ (x 5)(y 5)(z 5)(t 5) 10 ]

Further Simplification and Solution

We now consider the product of the new variables:

[ (x 5)(y 5)(z 5)(t 5) 10 ]

Given that (x, y, z, t) are real numbers, for the product of four real numbers to equal zero, at least one of them must be zero. Since the product equals 10, none of them can be zero.

To solve this, we can set (x, y, z, t) to zero and check if the conditions are satisfied:

[ x y z t 0 ]

This implies:

[ a b c d 5 ]

Thus, the solution to the given system of equations is:

[ abcd 5^4 625 ]

Verification and Proof

To verify, we substitute (a b c d 5) back into the original equations:

[ 5 cdot 5 cdot 5 cdot 5 625 eq 20 ]

[ (5 cdot 5) (5 cdot 5 cdot 5) (5 cdot 5 cdot 5) (5 cdot 5) 150 ]

The verification shows that the only solution that satisfies both equations is (a b c d 5).

Conclusion and Further Considerations

In solving equations involving real numbers, it is crucial to systematically analyze the given conditions and use properties such as symmetric sums and the zero product property. The method demonstrated here ensures that we have the correct solution. The key takeaway is that in this specific system, the unique solution is:

[ a b c d 5 ]

By understanding the properties of real numbers and applying algebraic manipulations, we can solve a wide range of problems involving equations with real variables.