Solving a System of Equations in Basic Calculus: An Alternative Approach

Introduction

This article provides an in-depth exploration of a system of equations in basic calculus, demonstrating an alternative method to solve it. The focus is on algebraic manipulation and the use of unique variables to derive a simplified solution. The content aims to offer clarity for those working with similar problems in calculus and to serve as a valuable reference for educational purposes.

The Problem Statement

The original problem involves a set of equations as follows:

abycz - ax  bczax - by  caxby - cz

Let us denote the left-hand side of each equation as k. Thus, we have:
k abycz - ax bczax - by caxby - cz

Deriving the Common Factor

ancor 'k' for each equation, we can express it as:

k/a bycz - ax (Equation 1)

k/b czax - by (Equation 2)

k/c axby - cz (Equation 3)

By combining terms, we get:

k(1/a)(1/b) cz (Equation 1')

k(1/a)(1/c) by (Equation 2')

k(1/b)(1/c) ax (Equation 3')

Solving for the Common Ratio

Now, let's solve for x, y, and z using the manipulated equations:

(1/bc)k x (from Equation 1')

(1/ac)k y (from Equation 2')

(1/ab)k z (from Equation 3')

From these, we can express the common ratio as:

x/bc y/ac z/ab

Conclusion and Proof

This solution demonstrates a method to solve a system of equations in basic calculus by utilizing algebraic manipulation. By introducing a common factor k and simplifying the equations, we arrive at a solution that shows the relationship between the variables x, y, and z.

The technique used here is an excellent example of problem-solving in calculus and can be applied to similar problems. The simplicity of the solution highlights the beauty and elegance of mathematical derivations.

References

This article is intended to provide an educational tool for understanding and solving problems in basic calculus. For further reading and more in-depth explanations, refer to standard calculus textbooks or online resources such as Khan Academy, MIT OpenCourseWare, and other reputable sources in calculus.