3x3 System Of Equations Solver With Work

3x3 System of Equations Solver with Work: A Comprehensive Guide

Solving a system of three equations with three unknowns (a 3x3 system) is a fundamental concept in algebra with wide-ranging applications in various fields like physics, engineering, economics, and computer science. While calculators and software can readily provide solutions, understanding the underlying methods is crucial for deeper comprehension and problem-solving abilities. This comprehensive guide will walk you through various methods for solving 3x3 systems, detailing the steps with illustrative examples and emphasizing the underlying mathematical principles.

Understanding 3x3 Systems of Equations

A 3x3 system of equations involves three linear equations, each containing three variables (typically represented as x, y, and z). 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.

Example:

Consider the following system:

• 2x + y - z = 5
• x - 2y + 3z = -4
• 3x + y + z = 8

Our aim is to find the values of x, y, and z that satisfy all three equations.

Methods for Solving 3x3 Systems

Several methods can effectively solve 3x3 systems. We will explore three of the most common:

1. Elimination Method (Gaussian Elimination)

This method systematically eliminates variables by adding or subtracting multiples of equations to create simpler equations. The process continues until a single variable is isolated, allowing for back-substitution to find the other variables.

Steps:

1. Choose a variable to eliminate: Select one variable and use two of the equations to eliminate it.
2. Eliminate the chosen variable from another pair of equations: Using the remaining equation and the result from step 1, eliminate the same variable again.
3. Solve for the remaining variable: You'll now have a single equation with one variable. Solve for this variable.
4. Back-substitute: Substitute the value found in step 3 into one of the equations from step 2 to find another variable.
5. Back-substitute again: Substitute the values found in steps 3 and 4 into one of the original equations to find the remaining variable.

Example using Elimination:

Let's solve the example system using elimination:

• Equation 1: 2x + y - z = 5
• Equation 2: x - 2y + 3z = -4
• Equation 3: 3x + y + z = 8

1. Eliminate 'x' from Equation 2 and Equation 3:

   • Multiply Equation 2 by -2: -2x + 4y - 6z = 8
   • Add this to Equation 1: (2x + y - z) + (-2x + 4y - 6z) = 5 + 8 => 5y - 7z = 13 (New Equation 4)
   • Multiply Equation 2 by -3: -3x + 6y - 9z = 12
   • Add this to Equation 3: (3x + y + z) + (-3x + 6y - 9z) = 8 + 12 => 7y - 8z = 20 (New Equation 5)

2. Eliminate 'y' from Equations 4 and 5:

   • Multiply Equation 4 by -7: -35y + 49z = -91
   • Multiply Equation 5 by 5: 35y - 40z = 100
   • Add these two equations: (-35y + 49z) + (35y - 40z) = -91 + 100 => 9z = 9 => z = 1

3. Back-substitute:

   • Substitute z = 1 into Equation 4: 5y - 7(1) = 13 => 5y = 20 => y = 4
   • Substitute z = 1 and y = 4 into Equation 1: 2x + 4 - 1 = 5 => 2x = 2 => x = 1

Therefore, the solution is x = 1, y = 4, z = 1.

2. Substitution Method

This method involves solving one equation for one variable and substituting the result into the other two equations. This process reduces the system to two equations with two unknowns, which can then be solved using similar techniques.

Steps:

1. Solve one equation for one variable: Choose an equation and solve it for one of the variables.
2. Substitute: Substitute the expression obtained in step 1 into the other two equations. This creates a system of two equations with two unknowns.
3. Solve the 2x2 system: Use either elimination or substitution to solve this smaller system.
4. Back-substitute: Substitute the values obtained in step 3 back into the equation from step 1 to find the value of the initially isolated variable.

Example using Substitution:

Let's revisit our example system using substitution:

• Equation 1: 2x + y - z = 5
• Equation 2: x - 2y + 3z = -4
• Equation 3: 3x + y + z = 8

1. Solve Equation 1 for y: y = 5 - 2x + z

2. Substitute: Substitute this expression for y into Equations 2 and 3:

   • Equation 2 becomes: x - 2(5 - 2x + z) + 3z = -4 => x - 10 + 4x - 2z + 3z = -4 => 5x + z = 6
   • Equation 3 becomes: 3x + (5 - 2x + z) + z = 8 => x + 2z = 3

3. Solve the 2x2 system: We now have:

   • 5x + z = 6
   • x + 2z = 3

   Multiply the second equation by -5: -5x - 10z = -15. Add this to the first equation: (-5x - 10z) + (5x + z) = -15 + 6 => -9z = -9 => z = 1

   Substitute z = 1 into x + 2z = 3: x + 2(1) = 3 => x = 1

4. Back-substitute: Substitute x = 1 and z = 1 into y = 5 - 2x + z: y = 5 - 2(1) + 1 => y = 4

Again, the solution is x = 1, y = 4, z = 1.

3. Cramer's Rule

Cramer's Rule is a method that uses determinants to solve systems of linear equations. It's particularly useful when dealing with systems of any size (not just 3x3), although it can become computationally intensive for larger systems.

Steps:

1. Calculate the determinant of the coefficient matrix: This matrix is formed by the coefficients of the variables in the equations.
2. Calculate the determinants of the matrices obtained by replacing each column of the coefficient matrix with the constant terms: For each variable, replace the column of its coefficients with the column vector of constants.
3. Solve for each variable: Divide the determinant obtained in step 2 (for each variable) by the determinant obtained in step 1.

Example using Cramer's Rule:

For our example system:

• 2x + y - z = 5
• x - 2y + 3z = -4
• 3x + y + z = 8

1. Coefficient Matrix and its Determinant:

   | 2  1 -1 |
   | 1 -2  3 |
   | 3  1  1 |
   

   The determinant is calculated as: 2(-2 - 3) - 1(1 - 9) + (-1)(1 + 6) = -10 + 8 - 7 = -9

2. Determinants with Constant Terms:

   • For x: Replace the first column with the constants:

     | 5  1 -1 |
     | -4 -2  3 |
     | 8  1  1 |
     

     The determinant is -9.

   • For y: Replace the second column with the constants:

     | 2  5 -1 |
     | 1 -4  3 |
     | 3  8  1 |
     

     The determinant is -36.

   • For z: Replace the third column with the constants:

     | 2  1  5 |
     | 1 -2 -4 |
     | 3  1  8 |
     

     The determinant is -9.

3. Solve for variables:

   • x = -9 / -9 = 1
   • y = -36 / -9 = 4
   • z = -9 / -9 = 1

The solution, once again, is x = 1, y = 4, z = 1.

Choosing the Right Method

The best method for solving a 3x3 system depends on the specific system and personal preference. Elimination is generally efficient and straightforward for most systems. Substitution can be advantageous when one equation easily solves for a single variable. Cramer's Rule offers a systematic approach but involves more calculations, making it less efficient for large systems. Understanding all three methods allows for flexibility and choosing the most appropriate approach depending on the context.

Applications of 3x3 Systems

Solving systems of equations is a cornerstone of many areas:

• Physics: Analyzing forces, motion, and circuit problems.
• Engineering: Designing structures, analyzing stresses, and modeling systems.
• Economics: Modeling supply and demand, and optimizing resource allocation.
• Computer Graphics: Transforming objects and creating realistic images.
• Computer Science: Solving linear programming problems and creating algorithms.

Understanding how to solve 3x3 systems is essential for effectively tackling problems in these and other fields. Mastering these techniques provides a strong foundation for more advanced mathematical concepts and problem-solving strategies. Practice is key; work through various examples and gradually increase the complexity of the systems you solve to build proficiency and confidence.