Solving a System of Linear Equations for Academic Success in Mathematics

Mathematics often requires students to tackle complex problems involving systems of equations. This article will walk you through the process of solving a system of three linear equations, illustrating the step-by-step approach to finding the solution. This is particularly useful for students looking to enhance their understanding of algebra and for professionals who need to solve such systems for various applications.

System of Equations Overview

Consider the following system of linear equations:

3x 4y 3z 9 x 4z 2 2x - 2y - 6z 4

To solve this system, we'll follow a structured approach to simplify and manipulate the equations to find the values of x, y, and z.

Step-by-Step Solution

Let's begin by simplifying the given equations:

3x 4y 3z 9 (1) x 4z 2 (2) 2x - 2y - 6z 4 (3)

Step 1: Multiply Equation (3) by 2

2(2x - 2y - 6z) 2(4)

4x - 4y - 12z 8 (4)

Step 2: Add Equation (1) and Equation (4)

(3x 4y 3z) (4x - 4y - 12z) 9 8

7x - 9z 17 (5)

Step 3: Multiply Equation (2) by 7

7(x 4z) 7(2)

7x 28z 14 (6)

Step 4: Subtract Equation (5) from Equation (6)

(7x 28z) - (7x - 9z) 14 - 17

37z -3

z -3/13

Step 5: Substitute z -3/13 into Equation (2)

x 4(-3/13) 2

x - 12/13 2

x 38/13

Step 6: Substitute x 38/13 and z -3/13 into Equation (1)

3(38/13) 4y 3(-3/13) 9

114/13 4y - 9/13 9

4y 9 - 105/13

4y 12/13

y 3/13

Hence, the solution is:

x 38/13, y 3/13, and z -3/13

Conclusion

In this detailed problem-solving exercise, we tackled a system of three linear equations and found the values of x, y, and z. This approach demonstrates crucial mathematical skills and is essential for anyone seeking to deepen their understanding of linear algebra.

For more resources and detailed explanations of such problems, visit the Google Mathematics Tutorials and Lessons section. Remember, practice is key to mastering these concepts!