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, including physics, engineering, economics, and computer science. While calculators and software can readily provide solutions, understanding the underlying methods is crucial for developing problem-solving skills and appreciating the mathematical logic involved. This comprehensive guide will walk you through several methods for solving 3x3 systems, providing detailed explanations and worked examples for each.

Understanding 3x3 Systems of Equations

A 3x3 system of equations consists of 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:

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

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

Methods for Solving 3x3 Systems

Several methods exist for solving 3x3 systems of equations. We'll explore three common and effective approaches:

1. Elimination Method (Gaussian Elimination)

The elimination method, also known as Gaussian elimination, is a systematic approach that involves manipulating the equations to eliminate variables one by one until a single variable remains. This method is particularly useful for systems with a unique solution.

Steps:

1. Choose a variable to eliminate: Select one variable and use two equations to eliminate it.
2. Eliminate the chosen variable from two pairs of equations: Perform elementary row operations (adding or subtracting multiples of equations) to eliminate the selected variable from two different pairs of equations.
3. Solve for the remaining two variables: You'll now have a 2x2 system, which can be solved using substitution or elimination.
4. Substitute and solve for the remaining variables: Substitute the solutions obtained in step 3 into one of the original equations to find the value of the eliminated variable.
5. Check your solution: Substitute the values of x, y, and z into all three original equations to verify the solution.

Worked Example (Elimination Method):

Let's solve the example system using elimination:

• x + y + z = 6 (Equation 1)
• 2x - y + z = 3 (Equation 2)
• x + 2y - z = 3 (Equation 3)

1. Eliminate z: Add Equation 1 and Equation 3: (x + y + z) + (x + 2y - z) = 6 + 3 => 2x + 3y = 9 (Equation 4)
2. Eliminate z (again): Subtract Equation 3 from Equation 2: (2x - y + z) - (x + 2y - z) = 3 - 3 => x - 3y = 0 (Equation 5)
3. Solve the 2x2 system: We now have Equations 4 and 5:
   • 2x + 3y = 9
   • x - 3y = 0 Adding these two equations gives 3x = 9, so x = 3. Substituting x = 3 into Equation 5: 3 - 3y = 0 => y = 1.
4. Solve for z: Substitute x = 3 and y = 1 into Equation 1: 3 + 1 + z = 6 => z = 2.
5. Check:
   • 3 + 1 + 2 = 6 (Correct)
   • 2(3) - 1 + 2 = 3 (Correct)
   • 3 + 2(1) - 2 = 3 (Correct)

Therefore, the solution is x = 3, y = 1, and z = 2.

2. Substitution Method

The substitution method involves solving one equation for one variable in terms of the other two, and then substituting this expression into the other two equations. This process reduces the system to a 2x2 system, which can then be solved further.

Steps:

1. Solve one equation for one variable: Choose one equation and solve it for one variable in terms of the other two.
2. Substitute: Substitute the expression obtained in step 1 into the other two equations.
3. Solve the resulting 2x2 system: Use elimination or substitution to solve the reduced 2x2 system.
4. Back-substitute: Substitute the values obtained in step 3 back into the equation from step 1 to find the value of the remaining variable.
5. Check your solution: Verify the solution by substituting the values into all three original equations.

Worked Example (Substitution Method): (This will be lengthy, to illustrate the process fully, but the elimination method is often more efficient for 3x3 systems) We'll use the same example system.

This method becomes cumbersome for 3x3 systems and is less efficient than elimination. It's best suited for smaller systems of equations. A full worked example using substitution for this 3x3 system would be excessively long and repetitive.

3. Cramer's Rule

Cramer's Rule provides a direct method for solving systems of linear equations using determinants. While elegant, it can be computationally intensive for larger systems.

Steps:

1. Find the determinant of the coefficient matrix: The coefficient matrix is formed by the coefficients of the variables in the system. If this determinant is zero, there is either no solution or infinitely many solutions.
2. Calculate the determinants of the modified matrices: Create three new matrices by replacing the column of coefficients for each variable with the constants on the right-hand side of the equations.
3. Solve for the variables: The solution is given by the ratio of determinants:
   • x = Det(x) / Det(A)
   • y = Det(y) / Det(A)
   • z = Det(z) / Det(A) where Det(A) is the determinant of the coefficient matrix, and Det(x), Det(y), and Det(z) are the determinants of the modified matrices.
4. Check your solution: Substitute the values into the original equations to verify the solution.

Worked Example (Cramer's Rule):

Again, using our example system:

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

1. Coefficient Matrix and Determinant:

The coefficient matrix is:

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

Calculating the determinant (using various methods such as cofactor expansion): Det(A) = -6

2. Modified Matrices and Determinants:

For Det(x), replace the first column with the constant vector (6, 3, 3):

| 6  1  1 |
| 3 -1  1 |
| 3  2 -1 |

Det(x) = -18

For Det(y), replace the second column with (6, 3, 3):

| 1  6  1 |
| 2  3  1 |
| 1  3 -1 |

Det(y) = -6

For Det(z), replace the third column with (6, 3, 3):

| 1  1  6 |
| 2 -1  3 |
| 1  2  3 |

Det(z) = -12

3. Solve for x, y, z:

• x = Det(x) / Det(A) = -18 / -6 = 3
• y = Det(y) / Det(A) = -6 / -6 = 1
• z = Det(z) / Det(A) = -12 / -6 = 2

4. Check: (Already done in the Elimination method example)

Therefore, the solution is again x = 3, y = 1, and z = 2.

Choosing the Right Method

The best method for solving a 3x3 system depends on the specific system and your preference. The elimination method is generally the most efficient and systematic for most cases, particularly when dealing with systems that have a unique solution. Cramer's rule is elegant but can be computationally expensive for larger systems. The substitution method is generally less efficient for 3x3 systems but can be useful in specific scenarios. Practice with each method will help you determine which approach works best for you.

Handling Systems with No Solution or Infinitely Many Solutions

Not all 3x3 systems have a unique solution. If, during the elimination or substitution process, you arrive at a contradiction (e.g., 0 = 1), the system has no solution (it is inconsistent). If you reach an identity (e.g., 0 = 0) that provides no further information, the system has infinitely many solutions (it is dependent). In these cases, Cramer's Rule will indicate this by producing a zero determinant for the coefficient matrix.

Applications of 3x3 Systems

Solving 3x3 systems of equations has many real-world applications. These include:

• Network analysis: Determining currents in electrical circuits.
• Engineering: Analyzing stresses and strains in structures.
• Physics: Solving problems involving forces and motion.
• Economics: Modeling market equilibrium.
• Computer graphics: Representing transformations and rotations in 3D space.
• Cryptography: Solving systems of equations in cryptography methods.

This comprehensive guide provides a solid foundation for understanding and solving 3x3 systems of equations. Remember that practice is key to mastering these techniques. Work through various examples, experimenting with different methods, to build your confidence and proficiency.