Solving Equations: From Simple Arithmetic to Complex Problem Solving

In mathematics, solving equations is an essential skill that bridges the concepts of basic arithmetic and more advanced algebra. This article will explore a specific problem involving arithmetic operations and algebraic manipulation, presenting a step-by-step solution to the equation: When a number multiplied by 3 and 7 gives the same result, and when the number is multiplied by 9 and then 12 is subtracted, what is the number?

Problem Statement

The problem can be stated as follows: Find a number x such that:

When the number x is multiplied by 3 and 7, the result is the same. When the number x is multiplied by 9 and then 12 is subtracted, the result is obtained.

Step-by-Step Solution

The conditions can be represented algebraically as:

3x 7x

9x - 12

Let's solve the first condition:

Solving the First Condition

3x 7x

Subtracting 3x from both sides gives:

0 4x

This implies that:

x 0

Evaluating the Second Condition with x 0

Now, let's evaluate the second condition with x 0:

90 - 12 0 - 12 -12

However, this approach seems to contradict the initial problem statement. Let's re-examine the conditions more carefully.

Revisiting the Conditions

We need to solve both conditions simultaneously:

3x 7x

9x - 12 4x - 12

The equation 3x 7x is not valid unless x is zero, but we need to consider the second condition as well. Let's simplify the second condition:

9x - 12 4x - 12

Subtracting 4x from both sides gives:

5x - 12 -12

Add 12 to both sides:

5x 0

This implies that:

x 0

Conclusion

The value of the number is 0. When multiplied by 9 and then 12 is subtracted, the result is -12.

Alternatively, another solution can be:

4x - 12 3x

Subtract 3x from each side:

x - 12 0

Add 12 to each side:

x 19

This also satisfies the conditions, as:

9 × 19 - 12 171 - 12 159

4 × 19 - 12 76 - 12 64

Thus, the number 19 is also a valid solution.

In summary, while the initial problem may seem to suggest a specific number, there are multiple valid solutions, primarily due to the nature of the arithmetic involved.

Key Takeaways:

Understanding the problem statement is crucial for solving equations. Check each solution against the problem conditions to ensure its validity. Practice different approaches to solving equations for a better grasp of the underlying concepts.

For further exploration, consider other arithmetic and algebraic problems, such as those involving percentages, ratios, and complex equations, to deepen your skills in problem-solving and mathematical reasoning.