How To Solve Systems With Three Variables

How to Solve Systems with Three Variables: A Comprehensive Guide

Solving systems of equations with three variables might seem daunting at first, but with a systematic approach and a solid understanding of the underlying principles, it becomes manageable and even enjoyable. This comprehensive guide will walk you through various methods, providing clear explanations and practical examples to help you master this crucial algebraic skill.

Understanding Systems of Three Variables

A system of three variables involves three equations, each containing three unknowns (typically represented as x, y, and z). The goal is to find the values of x, y, and z that satisfy all three equations simultaneously. These values represent the point of intersection in a three-dimensional coordinate system. Unlike two-variable systems which graphically represent lines, three-variable systems represent planes, and the solution is where these three planes intersect.

There are several ways a system of three variables can behave:

• Unique Solution: The three planes intersect at a single point. This means there's one unique solution (x, y, z) that satisfies all three equations.
• Infinite Solutions: The three planes intersect along a line, meaning there are infinitely many solutions.
• No Solution: The planes do not intersect at any common point. This could be because they are parallel, or they intersect in pairs but not all three at once.

Methods for Solving Systems of Three Variables

We'll explore three primary methods: elimination, substitution, and using matrices (with Gaussian elimination).

1. Elimination Method

The elimination method focuses on systematically eliminating one variable at a time until you have a single equation with one variable. Here's a step-by-step approach:

Step 1: Choose a variable to eliminate. Look for equations where a variable has the same coefficient (but opposite signs – this simplifies calculations) or where it's easy to make coefficients equal by multiplying equations.

Step 2: Eliminate the chosen variable. Add or subtract equations strategically to eliminate the chosen variable. This will result in two new equations with only two variables.

Step 3: Solve the system of two variables. Use either elimination or substitution to solve this smaller system. You'll obtain values for two of the variables.

Step 4: Substitute and solve for the remaining variable. Substitute the values obtained in Step 3 back into any of the original three equations to solve for the remaining variable.

Step 5: Check your solution. Substitute the values of x, y, and z into all three original equations to ensure they satisfy all equations.

Example:

Let's solve the following system:

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

Solution:

1. Eliminate z: Subtract equation (1) from equation (2) and equation (1) from equation (3):

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

2. Solve the system of two variables: We now have:

   • x - 2y = -3
   • y - 2z = -3

   Let's solve for x and y using the first equation: x = 2y - 3

3. Substitute: Substitute x = 2y - 3 into one of the original equations. Let's use equation (1):

   (2y - 3) + y + z = 6 => 3y + z = 9

4. Solve for z: We have a system of two variables now:

   • y - 2z = -3
   • 3y + z = 9

   Multiply the second equation by 2: 6y + 2z = 18. Add this to the first equation:

   7y = 15 => y = 15/7

5. Solve for other variables: Substitute y = 15/7 into y - 2z = -3 to find z. Then, substitute y and z into one of the original equations to solve for x.

6. Check your solution: Substitute the values of x, y, and z back into the original equations to verify the solution.

2. Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other two equations. This reduces the system to a system of two variables, which can then be solved using the same method or elimination.

Example:

Let's use the same system as before:

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

Solution:

1. Solve for one variable: Solve equation (1) for x: x = 6 - y - z

2. Substitute: Substitute this expression for x into equations (2) and (3):

   • 2(6 - y - z) - y + z = 3
   • (6 - y - z) + 2y - z = 3

3. Simplify and solve the system of two variables:

   • 12 - 2y - 2z - y + z = 3 => -3y - z = -9
   • 6 - y - z + 2y - z = 3 => y - 2z = -3

   Now solve this system of two variables (using either substitution or elimination) to find y and z.

4. Substitute back: Substitute the values of y and z into x = 6 - y - z to find x.

5. Check your solution: Verify the solution by substituting the values of x, y, and z into all three original equations.

3. Using Matrices (Gaussian Elimination)

Gaussian elimination is a powerful method for solving systems of linear equations, especially when dealing with larger systems. It involves manipulating a matrix representing the system using row operations to transform it into a row-echelon form, from which the solution can be easily read.

Step 1: Create the augmented matrix. The augmented matrix represents the coefficients of the variables and the constants.

Step 2: Perform row operations. Use elementary row operations (swapping rows, multiplying a row by a non-zero constant, adding a multiple of one row to another) to transform the matrix into row-echelon form (or reduced row-echelon form).

Step 3: Back-substitution. Once in row-echelon form, use back-substitution to solve for the variables.

Example (using the same system):

The augmented matrix is:

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

Row operations are performed to get it into row-echelon form. This involves a series of steps to systematically eliminate variables, resulting in a triangular matrix. The process can be quite involved and requires careful attention to detail. Once in row-echelon form, back-substitution is used to solve for the variables.

Choosing the Right Method

The best method for solving a system of three variables depends on the specific equations.

• Elimination: Works well when you can easily eliminate variables by adding or subtracting equations.
• Substitution: Useful when one variable is easily isolated in one of the equations.
• Matrices: Most efficient for larger systems or systems with more complicated coefficients. It's also a more systematic and less prone to errors approach, but requires a good understanding of matrix operations.

Handling Special Cases

Remember to consider special cases:

• Infinite Solutions: If during the elimination or substitution process, you arrive at an equation like 0 = 0, it indicates infinite solutions.
• No Solution: If you arrive at an equation like 0 = a non-zero number, it indicates no solution.

Mastering systems of three variables requires practice and patience. Start with simpler examples and gradually work your way up to more complex ones. By consistently applying these methods and understanding the underlying concepts, you'll be able to confidently solve any system of three variables. Remember to always check your solutions!