Solving Equations Through Logical Reasoning: A Step-by-Step Guide

Sometimes, solving a seemingly complex equation can be made much simpler through logical reasoning and a clear understanding of the operations involved. Consider the following problem:

If you multiply a number by 5, then divide by 5, add 8, and then subtract by 9, the result is 2. What is the number?

Breaking Down the Problem

Let's denote the unknown number as x. The problem statement can be described by setting up the following equation by following the operations step by step:

Multiply by 5: 5x Divide by 5: 5x/5 x Adding 8: x 8 Subtracting 9: x 8 - 9 x - 1

We know that the final result is 2, so we can write the equation:

x - 1 2

Solving for x

Now, solve the equation by adding 1 to both sides:

x - 1 1 2 1

This simplifies to:

x 3

Thus, the number is 3.

Additional Insights Using Algebra

We can also break down the problem algebraically:

Start with the equation: 5x/5 - 8 9 2 Perform division: x - 8 9 2 Simplify by subtracting 8 and adding 9 to one side: x 2 8 - 9 This simplifies to: x 1

However, we can additionally verify that:

5 * 3 / 5 8 - 9 2

To prove it, substitute x 3 into the original equation:

5 * 3 / 5 8 - 9 2

Which simplifies to:

15 / 5 8 - 9 2

And further simplifies to:

3 8 - 9 2

11 - 9 2

2 2

This confirms that the value of x is indeed 3.

Alternative Logical Approach

Another way to solve this problem is by recognizing that the multiplication and division by 5 cancel each other out. This leaves us with:

x 8 - 9 2

Which simplifies to:

x - 1 2

Adding 1 to both sides, we get:

x 3

Therefore, the number is 3.

Summary

Through logical reasoning and step-by-step algebraic simplification, we have determined that the number x is 3. This problem demonstrates the importance of understanding basic mathematical operations and their effects, as well as the value of recognizing patterns and simplifying equations to their core.