Solving Equations: A Step-by-Step Guide to Finding the Values of x and y

Algebra is a fundamental area of mathematics that involves the manipulation and solving of equations. In this guide, we will walk through the process of solving the given equations to find the values of x and y. The specific problem we are addressing is: Given xy 20 and x - y 1, what is the value of x?

Understanding the Problem

The problem involves two equations:

These equations provide a system of simultaneous equations that need to be solved to find the values of x and y.

Step 1: Express y in terms of x

To use the second equation, we can express y in terms of x:

y x - 1

Step 2: Substitute y in the first equation

Now, substitute the value of y from the second equation into the first equation:

x(x - 1) 20

This simplifies to:

x^2 - x - 20 0

Step 3: Solve the quadratic equation

The equation x^2 - x - 20 0 is a quadratic equation. To solve it, we can use the factoring method. First, we look for two numbers that multiply to -20 and add to -1. These numbers are -5 and 4.

Therefore, we can rewrite the equation as:

(x - 5)(x 4) 0

Setting each factor to zero gives us two possible solutions:

x - 5 0 or x 4 0

Solving these equations, we find:

x 5 or x -4

Step 4: Identify the valid solution

Given the context of the problem, we need to determine which value of x is valid. Let's consider both solutions.

Case 1: x 5

If x 5, then from the equation y x - 1, we have:

y 5 - 1 4

Checking the original equations:

This solution satisfies both equations.

Case 2: x -4

If x -4, then from the equation y x - 1, we have:

y -4 - 1 -5

Checking the original equations:

This solution also satisfies both equations.

Conclusion

Both solutions, x 5 and x -4, are valid within the context of the given equations. The problem does not specify additional constraints, so both values of x are correct answers.

By following these steps, you can systematically approach similar types of problems involving systems of equations in algebra. This process is essential in various fields, including physics, engineering, and economics, where understanding relationships between variables is crucial.