Solving Linear Equations: A Comprehensive Guide

Linear equations, as part of the vast realm of algebra, often pose intriguing challenges when solving for variables. This guide will delve into the details of solving a specific set of linear equations, namely, x2 y2 - 2xy 0 and x2 - xy y2 12. These equations present a unique puzzle that involves a blend of algebraic manipulation and the application of quadratic equations. Let's explore the step-by-step process to solve them.

Introduction to Linear and Quadratic Equations

Before diving into the solution, it's essential to understand the nature of the equations involved. Linear equations, such as x y 5, involve variables with a power of one. Quadratic equations, such as x2 - 5x 6 0, feature a variable with a power of two and represent parabolic graphs.

Solving the Given Linear Equations

Step 1: Simplify the First Equation

The first equation is x2 y2 - 2xy 0. To simplify this, we can rewrite it as:

x2 y2 - 2(x)(y) 0

This can be recognized as a perfect square trinomial:

(x - y)2 0

Therefore, we can conclude that:

x - y 0 x y

Step 2: Substitute and Solve the Second Equation

Now, we substitute y x into the second equation: x2 - xy y2 12.

Substituting:

x2 - x(x) (x)2 12 x2 - x2 x2 12 x2 12

Solving for x:

x ±√12 x ±2√3

Therefore, we have:

x 2√3 x -2√3

Since y x, the solutions for the system of equations are:

(x, y) (2√3, 2√3) (x, y) (-2√3, -2√3)

Understanding the Solution Process

In the process of solving these equations, several key algebraic concepts were employed:

Recognition of Perfect Square Trinomials: The first equation x2 y2 - 2xy 0 was simplified to (x - y)2 0, which helped in finding the relationship between x and y. Substitution: By substituting y x into the second equation, the complexity of the problem was reduced, leading to a simpler quadratic equation. Quadratic Solutions: Solving the quadratic equation x2 12 involved finding the square roots, which are the possible values for x.

Conclusion and Further Reading

Solving linear and quadratic equations is a fundamental skill in algebra. By understanding the processes and concepts involved, students can tackle a variety of problems with confidence. For those looking to further explore the subject, there are numerous resources and videos available online.

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Resources for Further Learning

Explore additional content through the following resources:

Khan Academy - Lessons on Linear Equations Math Is Fun - Quadratic Equations Explained Several Steps Interactive Videos - Step-by-Step Solving of Equations