Solving for X and Y in Given Equations: An In-Depth Guide

Mathematics often presents us with complex problems that require solving systems of equations. In this article, we will explore how to solve for X and Y in the given equations XY12 and 3X4Y1. This guide is designed to be thorough and clear, making it easy for anyone to follow and understand.

Introduction to the Problem

The problem at hand is to find the values of X and Y given the equations:

XY12 3X4Y1

Step-by-Step Solution

Step 1: Simplify the Second Equation

We start by simplifying the second equation 3X4Y1. Notice that the term 3X4Y can be written as 3X(4Y). This step will make the subsequent calculation easier to follow.

Step 2: Multiply the First Equation by 3

Next, we multiply the first equation XY12 by 3:

3XY36

Step 3: Subtract the Equations

We now subtract the modified first equation from the simplified second equation:

3XY36 (from the first equation multiplied by 3) 3X4Y1

Step 4: Solve for Y

By subtracting the second equation from the first, we get:

-Y36-135

Thus, we find:

Y-35

Step 5: Solve for X

With the value of Y, we can now solve for X using the first original equation:

X(-35)12

Solving for X, we get:

X12/(-35)-35/12≈-2.83 (rounded to two decimal places)

Step 6: Verification

It's important to verify the solution by substituting the values of X and Y back into the original equations:

For XY12: For 3X4Y1:

Both equations should hold true with the calculated values of X and Y.

Conclusion

In conclusion, we have successfully solved for X and Y using the given equations XY12 and 3X4Y1. The values of X and Y are approximately -2.83 and -35, respectively. This process demonstrates the power of algebraic manipulation and the importance of systematic problem solving in mathematics.

Frequently Asked Questions

What are simultaneous equations?

Simultaneous equations are a set of two or more equations that are considered true together. To solve such equations, we need to find the values of the variables that satisfy all the equations at the same time.

How can I check if my solution is correct?

Verification is key! Always substitute your solution back into the original equations to ensure they hold true. This step helps catch any potential errors in the calculation process.

Why is it important to manipulate equations correctly?

Correct manipulation ensures we arrive at the right solution. Missteps can lead to incorrect or unexpected results, making the process more difficult and less reliable.