Solving for a, b, and c: A Comprehensive Guide on Algebraic Manipulation
When dealing with algebraic equations, understanding the step-by-step process is crucial to finding the solution. This article provides a detailed walkthrough of solving three equations (Equation 1, Equation 2, and Equation 3) to find the values of a, b, and c. The equations to be solved are:
Equation 1: abc 12
Equation 2: ab - c 1
Equation 3: 2a - bc 8
Let's break down the process step-by-step to find the values of a, b, and c.
Step 1: Add Equation 1 and Equation 2
2ab 13
From this equation, we can find the value of ab by dividing both sides by 2:
ab 6.5
Step 2: Add Equation 2 and Equation 3
3a 9
By dividing both sides by 3, we can solve for a:
a 3
Step 3: Substitute the Value of a into Equation 3
Substituting a 3 into Equation 3:
2(3) - bc 8
6 - bc 8
-bc 2
bc 2
Step 4: Substitute the Value of a into Equation 1 and Equation 2
Substituting a 3 into Equation 2:
3b - c 1
Substituting a 3 and ab 6.5 into Equation 1:
3bc 12
bc 4
Since we already have bc 2 from the previous step, we now have a contradiction. Let's correct the process:
Revised Step 4: Corrected Equations and Values
From the corrected process, we have:
3a 9,
a 3
2b - c 6 - 8 -2,
2b 7,
b 3.5
abc 12
3bc 9,
bc 3
ab - c 1
ab 6.5
c 9 - 3.5 5.5
a 12 - 9 3
So, the values are:
Final Values:
a 3
b 3.5
c 5.5
These values satisfy all the given equations. The process demonstrates the importance of checking each step and ensuring that all variables are correctly substituted and solved.
Conclusion
Understanding and correctly solving algebraic equations is a fundamental skill in mathematics. This guide provides a clear and step-by-step breakdown of solving a system of equations to find the values of a, b, and c. The approach helps in verifying each step and ensures accurate results.
Further Reading
To delve deeper into algebra, consider reading about the solving equations method, linear algebra concepts, and the practice questions for a more comprehensive understanding.
Solving Equations
This section covers various techniques for solving algebraic equations, including substitution, elimination, and graphing. Understanding these methods is crucial for solving more complex problems.
Linear Algebra
Linear algebra extends the principles of algebra to higher dimensions, offering tools and techniques for solving systems of linear equations more efficiently. This section introduces matrix operations, vector spaces, and other foundational concepts.
Practice Questions
To reinforce the concepts learned, practice questions are essential. These questions cover a range of difficulty levels and help solidify your understanding of algebraic manipulation and problem-solving techniques.