Solving Mathematical Problems: A Step-by-Step Guide with Examples

Mathematics is a fundamental subject that plays a crucial role in our daily lives. From simple calculations to complex problem solving, understanding mathematical concepts is essential. In this article, we will explore how to solve a specific type of mathematical problem and provide step-by-step solutions. We will also introduce important mathematical terms and techniques. Let's get started.

Introduction to Mathematical Equations

A mathematical equation is a statement that asserts the equality of two expressions. In this context, we will be dealing with linear equations, which are equations of the first degree. Linear equations can be written in the form ax b c, where a, b, and c are constants and x is the variable. The goal is to find the value of the variable that makes the equation true.

Example Problem

Consider the following problem: I am thinking of a number. If I multiply it by 6, then add -4, and then divide by 2, the result is 25. What is the number?

Solution 1

The equation can be written as follows:

[ frac{6x - 4}{2} 25 ]

First, we multiply both sides of the equation by 2 to eliminate the fraction:

[ 6x - 4 50 ]

Next, we add 4 to both sides:

[ 6x 54 ]

Finally, we divide both sides by 6:

[ x 9 ]

Thus, the number you are thinking of is 9.

Solution 2

Let the number be x. According to the problem, the equation can be set up as:

[ frac{6x - 4}{2} 25 ]

Positing the order of operations, we first multiply 6x by 2:

[ 6x - 2 25 ]

Next, we solve for x by adding 2 to both sides:

[ 6x 27 ]

Finally, we divide both sides by 6:

[ x frac{27}{6} 4.5 ]

To verify the solution, we substitute x 4.5 back into the original equation:

[ frac{6(4.5) - 4}{2} frac{27 - 4}{2} frac{23}{2} 11.5 eq 25 ]

There seems to be a mistake in the verification process. Let's recheck the solution:

Let x 9

Substituting into the original equation:

[ frac{6(9) - 4}{2} frac{54 - 4}{2} frac{50}{2} 25 ]

Thus, the number is indeed 9.

More Examples

Example 3

Suppose the number is 9. If multiplied by 6, it becomes 54. Subtracting 4 gives 50. Dividing by 2 results in 25.

So, the number is proved to be 9.

Example 4

Let the number be x. According to the problem, the equation can be set up as:

[ 6x - 4 25 ]

Add 4 to both sides:

[ 6x 29 ]

Divide both sides by 6:

[ x frac{29}{6} approx 4.833 ]

Thus, the number is approximately 4.833.

Conclusion

In conclusion, solving mathematical problems, especially linear equations, involves a step-by-step approach. By understanding and practicing these techniques, one can efficiently solve various problems. The key is to meticulously follow each step and verify the solution.

For more practice and to improve your mathematical skills, consider working through similar problems and continuously refining your problem-solving techniques. Happy math solving!