Solving Systems of Equations Involving Multiplication: A Comprehensive Guide

Understanding and solving systems of equations with multiplication is a crucial skill in algebra. These problems often involve finding the values of variables given multiple algebraic expressions. This article will explore the methods and strategies to solve such systems, using detailed examples to illustrate the process.

Introduction to Systems of Equations Involving Multiplication

A system of equations involving multiplication includes expressions where the variables are multiplied together. These types of systems are commonly found in real-world problems, such as balancing chemical equations or solving economic models. To solve such systems, we often use substitution, elimination, or a combination of both methods.

Example Problem

Consider the following system of equations:

a × b 2 b × c 24 c × a 3

Our goal is to find the values of a, b, and c that satisfy all three equations simultaneously. Let's begin by expressing each variable in terms of the others.

Step 1: Express b in terms of a

b 2 a

Step 2: Express c in terms of a

2 a × c 24 c 24 times; a 2 12 a

Step 3: Express c in terms of a and substitute

12 a × a 3 12 a 2 3 a 2 3 12 1 4 a 1 4 1 2

Step 4: Find b and c

b 2 1 2 4 c 12 × 1 2 6

Step 5: Calculate the product a × b × c

1 2 × 4 × 6 1 2 × 24 10.5

Conclusion

Through careful substitution and algebraic manipulation, we have successfully found the values of a, b, and c that satisfy the given system of equations. This problem highlights the importance of methodical problem-solving in mathematics, particularly in handling systems of equations involving multiplication.

Related Keywords

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Further Reading and Practice

For further practice and a deeper understanding of solving systems of equations involving multiplication, consider the following resources:

Khan Academy - Systems of Equations with Multiplication Math is Fun - Systems of Equations