Solving Systems of Linear Equations for Variables a, b, and c

Given the following system of linear equations:

2a 3b 4c 35

3a 5b 7c 30

We will explore step-by-step methods to find the values of a, b, and c. This method is crucial for understanding algebraic methods, especially in solving systems of linear equations. It involves elimination, substitution, and matrix methods. This content is valuable for students and professionals in STEM fields and data analysis.

Step-by-Step Solution

Step 1: Eliminate One Variable

We start by eliminating one of the variables. Let's eliminate c.

Multiply the first equation by 7 and the second equation by 4:

7(2a 3b 4c) 7(35) rarr; 14a 21b 28c 245 (Equation 3) 4(3a 5b 7c) 4(30) rarr; 12a 20b 28c 120 (Equation 4)

Step 2: Subtract the Equations

Subtract Equation 4 from Equation 3:

14a 21b 28c - (12a 20b 28c) 245 - 120

14a 21b 28c - 12a - 20b - 28c 125

2a b 125 (Equation 5)

Step 3: Solve for One Variable

Express b in terms of a:

b 125 - 2a (Equation 6)

Step 4: Substitute Back to Find c

Substitute Equation 6 into the first original equation:

2a 3(125 - 2a) 4c 35

2a 375 - 6a 4c 35

-4a 4c 35 - 375

-4a 4c -340

c a - 85 (Equation 7)

Step 5: Substitute c Back

Substitute Equations 6 and 7 into a, b, and c:

a b c a (125 - 2a) (a - 85)

a 125 - 2a a - 85 40

40 40

Hence, the value of a, b, and c is:

boxed{40}

Alternative Methods

Another approach involves using matrix methods and Gaussian elimination. Here's an overview of the process:

Define abc k as a variable. Set up the linear system: 2a 3b 4c 35 3a 5b 7c 30 Create the augmented matrix: left[ begin{array}{cccc} 1 1 1 k 2 3 4 35 3 5 7 30 end{array} right] Apply row operations: R_2 - 2R_1 rarr; left[ begin{array}{cccc} 1 1 1 k 0 1 2 35 - 2k 0 2 4 30 - 3k end{array} right] R_3 - 2R_1 rarr; left[ begin{array}{cccc} 1 1 1 k 0 1 2 35 - 2k 0 0 0 k - 40 end{array} right] The system has a solution if and only if k 40.

These methods are typically covered in advanced algebra courses and are essential for understanding the underlying principles of solving systems of linear equations.

To learn more about solving systems of linear equations and algebraic methods, we recommend visiting MathIsFun and Khan Academy.

Remember, mastering these methods is crucial for advanced mathematics and its applications in various fields such as engineering, physics, and computer science.