Solving Equations with the Elimination Method - A Case Study with 2A3B and 2A - 3B

When it comes to solving algebraic equations, the elimination method is a powerful tool. In this article, we will explore how to solve the problem of 2A3B 10 and 2A - 3B 15 using this method. Let's break down the process step by step and understand the underlying principles.

Introduction to the Problem

Consider the two equations:

2A3B 10 2A - 3B 15

We will solve these equations using the elimination method, which involves adding or subtracting the equations to eliminate one of the variables.

Step-by-Step Solution

Step 1: Labeling the Equations

First, we label the two equations as follows:

Equation 1: 2A3B 10 Equation 2: 2A - 3B 15

Step 2: Using the Elimination Method

The goal is to eliminate one of the variables. We can achieve this by adding or subtracting the equations. Let's first look at the coefficients of B in both equations. Notice that the coefficients are the same in magnitude but opposite in sign: 3B and -3B.

Adding the two equations will eliminate the B variable:

Equation 1: 2A3B 10 Equation 2: 2A - 3B 15 3A 25

From the resulting equation, we can solve for A:

3A  25
A  25 / 3

Step 3: Substituting the Value of A

Now that we have the value for A, we can substitute it back into one of the original equations to solve for B. Let's use Equation 2:

2A - 3B 15

Substitute A 25 / 3 into the equation:

2(25/3) - 3B  15
50/3 - 3B  15
50/3 - 15  3B

Simplify the equation:

(50 - 45) / 3  3B
5 / 3  3B
B  5 / 9

Therefore, we have:

A  25 / 3
B  5 / 9

Conclusion

With the elimination method, we have successfully solved the algebraic problem 2A3B 10 and 2A - 3B 15. By carefully adding or subtracting the equations, we can eliminate one variable, simplify the problem, and solve for the other variable. This method is a fundamental tool in algebra and can be applied to a wide range of problems.

Understanding the elimination method not only enhances problem-solving skills but also provides a deeper insight into the structure of algebraic equations. Whether you are a student, a teacher, or a professional working with mathematical models, mastering this technique will greatly enhance your ability to tackle complex algebraic problems.

Key Takeaways

The elimination method is a powerful tool for solving systems of equations. Identify and manipulate the coefficients of the variables to eliminate one of them. Substitute the found value back into the original equations to solve for the other variable.

Further Exploration

Explore other algebraic problems and see how the elimination method can be applied. Practice with a variety of equations to build confidence and proficiency in this essential skill.