Solving A 3x3 System Of Linear Equations

Solving a 3x3 System of Linear Equations: A Comprehensive Guide

A 3x3 system of linear equations involves three equations with three unknowns, typically represented as x, y, and z. Solving these systems is a fundamental concept in algebra with applications spanning various fields, from engineering and physics to computer science and economics. This comprehensive guide will explore different methods to solve these systems, from the elementary elimination method to the more sophisticated matrix methods, explaining each step clearly and providing examples for better understanding. We'll also address common challenges and misconceptions.

Introduction to 3x3 Linear Systems

Before diving into the methods, let's understand what a 3x3 system looks like:

a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃

Here, a₁, b₁, c₁, d₁, etc., are constants, and x, y, z are the variables we need to solve for. The goal is to find the values of x, y, and z that simultaneously satisfy all three equations. A system can have one unique solution, infinitely many solutions, or no solution at all.

Method 1: Elimination Method (Gaussian Elimination)

This method systematically eliminates variables until we obtain a single equation with one unknown. We can then solve for that unknown and substitute back into previous equations to find the others. Let’s illustrate with an example:

Example:

Solve the following system:

x + y + z = 6
2x - y + z = 3
x + 2y - z = 3

Steps:

1. Eliminate one variable: Let's eliminate z from the first two equations. Subtract the first equation from the second:

   (2x - y + z) - (x + y + z) = 3 - 6 which simplifies to x - 2y = -3 (Equation 4)

2. Eliminate the same variable from another pair: Now, let's eliminate z from the first and third equations. Add the first and third equations:

   (x + y + z) + (x + 2y - z) = 6 + 3 which simplifies to 2x + 3y = 9 (Equation 5)

3. Solve the resulting 2x2 system: We now have a system with two equations and two unknowns (x and y):

   x - 2y = -3 (Equation 4) 2x + 3y = 9 (Equation 5)

   We can solve this using elimination again. Multiply Equation 4 by 2: 2x - 4y = -6. Subtract this from Equation 5:

   (2x + 3y) - (2x - 4y) = 9 - (-6) which simplifies to 7y = 15, so y = 15/7

4. Substitute back: Substitute y = 15/7 into Equation 4: x - 2(15/7) = -3, which gives x = -3 + 30/7 = 9/7

5. Substitute to find the remaining variable: Substitute the values of x and y into the original first equation:

   (9/7) + (15/7) + z = 6 which gives z = 6 - 24/7 = 18/7

Solution: Therefore, the solution to the system is x = 9/7, y = 15/7, z = 18/7.

Method 2: Matrix Method (Gaussian Elimination with Matrices)

This method represents the system of equations as an augmented matrix and uses row operations to transform it into row-echelon form or reduced row-echelon form.

Example using the same system as above:

The augmented matrix is:

[ 1  1  1 | 6 ]
[ 2 -1  1 | 3 ]
[ 1  2 -1 | 3 ]

Row Operations:

1. Subtract 2 times the first row from the second row:

   [ 1  1  1 | 6 ]
   [ 0 -3 -1 |-9 ]
   [ 1  2 -1 | 3 ]
   

2. Subtract the first row from the third row:

   [ 1  1  1 | 6 ]
   [ 0 -3 -1 |-9 ]
   [ 0  1 -2 |-3 ]
   

3. Divide the second row by -3:

   [ 1  1  1 | 6 ]
   [ 0  1 1/3 | 3 ]
   [ 0  1 -2 |-3 ]
   

4. Subtract the second row from the third row:

   [ 1  1  1 | 6 ]
   [ 0  1 1/3 | 3 ]
   [ 0  0 -7/3 |-6 ]
   

5. Multiply the third row by -3/7:

   [ 1  1  1 | 6 ]
   [ 0  1 1/3 | 3 ]
   [ 0  0  1 | 18/7 ]
   

This is now in row-echelon form. We can now use back-substitution to solve for x, y, and z, just like in the elimination method, leading to the same solution as before. Reduced row-echelon form would further simplify the matrix to a diagonal matrix, making the solution even more direct.

Method 3: Cramer's Rule

Cramer's rule is a method that uses determinants to solve for the unknowns. For a 3x3 system, it involves calculating several 3x3 determinants. While elegant, it can be computationally intensive for larger systems.

Example (using the same system):

1. Calculate the determinant of the coefficient matrix (D):

   D = | 1  1  1 |
       | 2 -1  1 |
       | 1  2 -1 | 
   

   D = 1(1 - 2) - 1(-2 - 1) + 1(4 + 1) = -1 + 3 + 5 = 7

2. Calculate the determinant for x (Dₓ): Replace the first column of the coefficient matrix with the constants:

   Dₓ = | 6  1  1 |
        | 3 -1  1 |
        | 3  2 -1 |
   

   Dₓ = 6(1 - 2) - 1(-3 - 3) + 1(6 + 3) = -6 + 6 + 9 = 9

3. Calculate the determinant for y (Dᵧ): Replace the second column with the constants:

   Dᵧ = | 1  6  1 |
        | 2  3  1 |
        | 1  3 -1 |
   

   Dᵧ = 1(-3 - 3) - 6(-2 - 1) + 1(6 - 3) = -6 + 18 + 3 = 15

4. Calculate the determinant for z (D₂): Replace the third column with the constants:

   D₂ = | 1  1  6 |
        | 2 -1  3 |
        | 1  2  3 |
   

   D₂ = 1(-3 - 6) - 1(6 - 3) + 6(4 + 1) = -9 - 3 + 30 = 18

5. Solve for x, y, and z:

   x = Dₓ / D = 9 / 7 y = Dᵧ / D = 15 / 7 z = D₂ / D = 18 / 7

This gives the same solution as before.

Inconsistent and Dependent Systems

Not all 3x3 systems have a unique solution. An inconsistent system has no solution. This often occurs when attempting to solve the system and encountering a contradiction, such as 0 = 1. A dependent system has infinitely many solutions. This happens when one equation is a multiple of another, leading to redundant information. In matrix form, this manifests as a row of zeros in the reduced row-echelon form.

Applications of 3x3 Linear Systems

3x3 systems are used extensively in various fields:

• Engineering: Analyzing forces in structural systems, circuit analysis.
• Physics: Solving problems involving Newton's laws, projectile motion.
• Computer Graphics: Representing transformations (rotation, scaling, translation) in 3D space.
• Economics: Modeling supply and demand, input-output analysis.

Frequently Asked Questions (FAQ)

Q: Which method is the best for solving 3x3 systems?

A: There's no single "best" method. Gaussian elimination (either the standard elimination method or the matrix version) is generally preferred for its efficiency and broad applicability. Cramer's rule is elegant but can become computationally expensive for larger systems.

Q: What if I get a solution where one variable is zero?

A: That's perfectly valid. A zero value for a variable is a legitimate part of the solution.

Q: What if I make a mistake during the elimination process?

A: Carefully check your work at each step. A small arithmetic error can lead to an incorrect solution.

Conclusion

Solving a 3x3 system of linear equations is a crucial skill in mathematics and its applications. Understanding the different methods—elimination, matrix methods, and Cramer's rule—allows you to choose the most appropriate approach depending on the specific system and the tools at your disposal. Practicing these methods with various examples is essential to mastering them. Remember to always check your solutions by substituting them back into the original equations to verify their validity. With consistent practice, you'll gain confidence and proficiency in solving these systems.