Solving the Equation X2y 6

Introduction

The equation X2y 6 is a simple algebraic equation that can be solved in multiple ways. This article will guide you through different methods to find the solutions to this equation, and explore the geometric interpretation of these solutions in the xy-plane.

Two Methods to Solve for Solutions

There are two main methods to solve the equation X2y 6 for different values of x or y. Let's consider both approaches:

Method 1: Direct Substitution

In this method, we choose specific values for x and solve for y, or vice versa.

Solution 1: Choose x 2

22y  64y  6y  6 / 4y  1.5 or 3/2

So one solution is (2, 3/2).

Solution 2: Choose x 0

02y  60  6 (This is not a valid solution as 0 cannot equal 6)

However, if we correct the equation to a valid form, we get:

02y  6y  6 / 0 (Undefined)

Instead, we should correct it to:

02y  62y  6y  3

So another solution is (0, 3).

Method 2: Parametric Equations

We can also use a parameter t to generate an infinite number of solutions. Let's consider the parametric forms:

Using t 1, 2, 3, ...

x  22t and y  2 - t

For t 1:

x  22 * 1  22, y  2 - 1  1Solution: (22, 1)

For t 2:

x  22 * 2  44, y  2 - 2  0Solution: (44, 0)

For t 3:

x  22 * 3  66, y  2 - 3  -1Solution: (66, -1)

These solutions demonstrate that there are infinitely many points (x, y) that satisfy the equation.

Geometric Interpretation

The equation X2y 6 can be interpreted as a curve in the xy-plane. The two solutions we found earlier, (2, 3/2) and (0, 3), are the points where this curve intersects the axes.

Y-Intercept

To find the y-intercept, set x 0:

02y  62y  6y  3

So the y-intercept is (0, 3).

X-Intercept

To find the x-intercept, set y 0:

x2 * 0  6x  0 (This is not a valid solution as 0 cannot equal 6)

However, if we correct the equation to a valid form, we get:

x2 * y  6x2 * 0  6x  6 / 0 (Undefined)

Instead, we should correct it to:

x2 * 0  6x  6 / 2  6

So the x-intercept is (6, 0).

Conclusion

By choosing different values for x or y, we can generate an infinite number of solutions to the equation X2y 6. The solutions (2, 3/2) and (0, 3) represent the points where this curve intersects the axes, and can be used to graph the equation in the xy-plane.