Solving M + 3N = 5 And M - N = 1 Find M Times N

Jul 12, 2025 by Admin 48 views

Solving a System of Equations and Finding the Product of Variables

Introduction

This article delves into solving a system of linear equations and subsequently calculating the product of the variables involved. The specific system we will address is:

M + 3N = 5
M - N = 1

Our objective is to determine the values of M and N that satisfy both equations simultaneously and then compute the product of these values (M × N). This problem falls under the category of algebra, a fundamental branch of mathematics that deals with symbols and the rules for manipulating those symbols. Solving systems of equations is a crucial skill in various fields, including engineering, economics, and computer science. This detailed exploration will guide you through the steps required to solve this system efficiently and accurately.

Methods for Solving Systems of Equations

There are several methods to solve systems of linear equations, including substitution, elimination, and graphical methods. Each method has its advantages, and the choice of method often depends on the specific equations involved. For this particular system, we will demonstrate the substitution method and the elimination method. These methods are particularly well-suited for systems with two variables and two equations, as they provide a structured approach to isolate and determine the values of the variables. Understanding these methods equips one with valuable problem-solving skills applicable in diverse mathematical contexts.

1. Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with a single variable, which can then be easily solved. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. This step-by-step approach ensures a clear and logical solution path.

2. Elimination Method

The elimination method, also known as the addition or subtraction method, involves manipulating the equations so that the coefficients of one of the variables are opposites or equal. By adding or subtracting the equations, we can eliminate one variable, resulting in a single equation with a single variable. This method is especially effective when the coefficients of one variable are easily made opposites or equal through multiplication. The elimination method provides a systematic way to simplify the system and solve for the variables.

Step-by-Step Solution

Using the Elimination Method

Let's solve the system of equations using the elimination method.

Write down the equations:

M + 3N = 5   (Equation 1)
M - N = 1     (Equation 2)

Subtract Equation 2 from Equation 1 to eliminate M:

(M + 3N) - (M - N) = 5 - 1
M + 3N - M + N = 4
4N = 4

Solve for N:
```
N = 4 / 4
N = 1
```
Substitute the value of N back into Equation 2 (or Equation 1) to solve for M:
```
M - N = 1
M - 1 = 1
M = 1 + 1
M = 2
```

Thus, we have found that M = 2 and N = 1 using the elimination method. This approach demonstrates the power of strategic manipulation to simplify and solve complex systems.

Using the Substitution Method

Alternatively, we can solve the system using the substitution method.

Solve Equation 2 for M:
```
M - N = 1
M = N + 1
```
Substitute the expression for M into Equation 1:
```
M + 3N = 5
(N + 1) + 3N = 5
```

Simplify and solve for N:

N + 1 + 3N = 5
4N + 1 = 5
4N = 5 - 1
4N = 4
N = 4 / 4
N = 1

Substitute the value of N back into the expression for M:
```
M = N + 1
M = 1 + 1
M = 2
```

Again, we find that M = 2 and N = 1. The substitution method provides a flexible alternative, reinforcing the solution obtained through elimination. Both methods converge on the same solution, highlighting the consistency of algebraic principles.

Calculating M × N

Now that we have found the values of M and N, we can calculate their product.

M = 2
N = 1

The product M × N is:

M × N = 2 × 1 = 2

Therefore, the product of M and N is 2. This final calculation provides the answer to the original question, completing the problem-solving process. The ability to accurately compute such products is crucial in various mathematical and real-world applications.

Verification

To ensure our solution is correct, we can substitute the values of M and N back into the original equations:

Equation 1: M + 3N = 5
```
2 + 3(1) = 2 + 3 = 5  (Correct)
```
Equation 2: M - N = 1
```
2 - 1 = 1  (Correct)
```

Since the values M = 2 and N = 1 satisfy both equations, our solution is verified. This step is essential in mathematical problem-solving, as it confirms the accuracy of the calculations and the validity of the results. Verification provides confidence in the solution and underscores the importance of precision in mathematical work.

Conclusion

In this article, we successfully solved the system of equations:

M + 3N = 5
M - N = 1

We found that M = 2 and N = 1, and their product, M × N, is 2. We used both the elimination and substitution methods to arrive at the same solution, demonstrating the consistency of these algebraic techniques. The solution was also verified by substituting the values back into the original equations, ensuring accuracy. Mastering these techniques is crucial for solving more complex mathematical problems and for applications in various scientific and engineering fields. Understanding these concepts builds a strong foundation for further mathematical studies.

This exercise highlights the importance of understanding and applying different methods to solve systems of equations. The ability to choose the most efficient method and to verify the solution are crucial skills in mathematics. By practicing these techniques, one can develop a strong foundation for tackling more complex problems in algebra and beyond. The journey through solving systems of equations is not just about finding answers but also about honing problem-solving skills that are valuable in various aspects of life.