Solving Simultaneous Equations with Working Out: ax by c and bx ay c

Introduction

Solving simultaneous equations is a fundamental skill in algebra, widely used in various fields from engineering to economics. This article focuses on solving the simultaneous equations ax by c and bx ay c. We'll explore the method step by step, including detailed explanations and examples.

Understanding the Equations

Let's consider the system of equations:

ax by c
bx ay c

To properly solve this system, it's crucial to recognize that these equations share the same constant term c. This inclusion can simplify the process of solving for x and y.

Methodology

We will proceed with a step-by-step approach to solve these simultaneous equations, considering different cases based on the values of a and b.

Scenario 1: a b

Let's start with the case where a b: a(x - y) 0 (a - b)(x - y) 0 Since a b, the term (a - b) is 0: "x - y 0"
x y Now, substituting x y into one of the original equations, say ax by c: a(x) b(x) c
(a b)x c Here are the possible solutions: If a b 0, and c ≠ 0, there is no solution. If a b ≠ 0 or a b c 0, there are infinitely many solutions.

Scenario 2: a ≠ b

If a ≠ b, we can create a simpler equation by subtracting one equation from the other: (ax by) - (bx ay) c - c
a(x - y) - b(x - y) 0
(a - b)(x - y) 0 Since a ≠ b, the term (a - b) is non-zero: "x - y 0"
x y With x y, we can substitute back into one of the original equations, such as ax by c: a(x) a(x) c
2ax c
x frac{c}{2a}
y frac{c}{2a} We have a unique solution for x and y unless 2a 0, meaning a 0 and c ≠ 0, which results in no solution.

Matrix Determinant Approach

For a more generalized approach, we can employ the determinant of the coefficient matrix to determine the uniqueness of the solution. Let us represent the system as:

A begin{bmatrix} a b b a end{bmatrix}
|A| a^2 - b^2

The determinant of the matrix is |A| a^2 - b^2. For the system to have a unique solution, the determinant must be non-zero:

If |A| a^2 - b^2 ≠ 0, the system has a unique solution.

Otherwise, if |A| a^2 - b^2 0, the system has either no solution or infinitely many solutions.

Conclusion

This methodical approach helps in identifying the solutions to the simultaneous equations ax by c and bx ay c under various conditions. The cases of a b and a ≠ b provide distinct paths to the solution, while the determinant approach offers a comprehensive understanding of the solution space.

References

1. Cramer's Rule (Wikipedia) 2. Matrix Determinant (Wolfram MathWorld)