Solving Equations and Finding Ratio: A Comprehensive Guide

When dealing with algebraic equations, it's often necessary to solve for certain variables and then find ratios or other values based on the solved equations. In this article, we will walk through the process of solving a system of equations and find the ratio x/y and explore the methods used in the process.

Introduction to the Problem

We are given two equations:

x y 3 (Equation 1)

x - y 1 (Equation 2)

We need to find the value of x/y .

Solving the System of Equations

First, let's add the two equations to eliminate y:

x y x - y 3 1

This simplifies to:

2x 4

Dividing both sides by 2:

x 2

Substitution and Finding y

Now that we have the value of x, we can substitute it back into one of the original equations. Let's use Equation 1:

2 y 3

Solving for y:

y 3 - 2 1

Final Ratio Calculation

Now that we have the values of x and y, we can calculate the desired ratio x/y :

2/1

This simplifies to:

Alternative Methods to Solve the Problem

Another approach to solving the same equations involves isolating one of the variables and substituting it into the other equation. Let's demonstrate using the second equation.

Isolating x and Substitution

From Equation 2, we have:

x - y 1

Adding y to both sides:

x 1 y

Substituting this into Equation 1:

1 y y 3

Simplifying:

1 2y 3

Subtracting 1 from both sides:

2y 2

Dividing both sides by 2:

y 1

Substituting y back into x:

x 1 1 2

Finally, the ratio x/y is:

2/1

Which again simplifies to 2.

Conclusion

In this article, we explored two different methods to solve a pair of linear equations and find the ratio of x/y. Both methods lead to the same result: 2. Understanding these techniques is crucial for tackling more complex algebraic problems and is a valuable skill in many fields, including mathematics, physics, and engineering.