How To Solve System Of Equations 3 Variables

How to Solve a System of Equations with Three Variables

Solving systems of equations with three variables might seem daunting, but with a systematic approach and a solid understanding of the underlying principles, it becomes a manageable task. This comprehensive guide will walk you through various methods, providing clear explanations and practical examples to help you master this essential algebraic skill. We’ll cover substitution, elimination, and using matrices, equipping you with the tools to tackle any three-variable system you encounter.

Understanding Systems of Equations with Three Variables

A system of equations with 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 space. Unlike two-variable systems which graphically represent lines, three-variable systems represent planes, and the solution represents where these three planes intersect.

Example:

Consider the following system:

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

This system represents three planes in three-dimensional space. Our objective is to find the (x, y, z) coordinates where these three planes intersect.

Method 1: Elimination Method

The elimination method, also known as the addition method, involves systematically eliminating one variable at a time by adding or subtracting equations. This process continues until you have a single equation with one variable, which can be easily solved.

Steps:

1. Choose two equations and eliminate one variable: Select any two equations from the system and manipulate them (multiplying by constants if necessary) so that when you add or subtract them, one variable cancels out.

2. Repeat the process with a different pair of equations: Use a different pair of equations from the original system and eliminate the same variable you eliminated in step 1. This will leave you with two equations in two variables.

3. Solve the resulting system of two equations: Use either elimination or substitution to solve this smaller system for the remaining two variables.

4. Substitute the values back into one of the original equations: Substitute the values you found in step 3 back into any of the original three equations to solve for the third variable.

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

Example using Elimination:

Let's solve the system we introduced earlier:

• 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: Now we have Equations 4 and 5: 2x + 3y = 9 x - 3y = 0

   Add Equation 4 and Equation 5: 3x = 9 x = 3

   Substitute x = 3 into Equation 5: 3 - 3y = 0 3y = 3 y = 1

4. Solve for z: Substitute x = 3 and y = 1 into Equation 1: 3 + 1 + z = 6 z = 2

5. Check: Equation 1: 3 + 1 + 2 = 6 (True) Equation 2: 2(3) - 1 + 2 = 7 (False - there's a mistake!)

Let's re-examine step 2. There was an error in the subtraction. It should be:

2. Eliminate z (again): Subtract Equation 3 from Equation 2: (2x - y + z) - (x + 2y - z) = 3 - 3 x - 3y + 2z = 0 (Equation 5 - Corrected)

Now we need a different approach to solve this. Let's use Equation 1 and Equation 2 to eliminate y:

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

Adding these two equations gives:

3x + 2z = 9 (Equation 6)

Now let's use Equation 1 and Equation 3 to eliminate y:

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

Multiply Equation 1 by 2: 2x + 2y + 2z = 12. Subtract Equation 3 from this: x + 3z = 9 (Equation 7)

Now solve the system formed by Equation 6 and Equation 7:

3x + 2z = 9 x + 3z = 9

Multiply Equation 7 by 3: 3x + 9z = 27 Subtract Equation 6 from this: 7z = 18; z = 18/7

Substitute z = 18/7 into Equation 7: x + 3(18/7) = 9; x = 9 - 54/7 = 9/7

Substitute x = 9/7 and z = 18/7 into Equation 1: 9/7 + y + 18/7 = 6; y = 6 - 27/7 = 15/7

Therefore the correct solution is (9/7, 15/7, 18/7)

This example highlights the importance of careful calculation in the elimination method. One small error can lead to an incorrect solution.

Method 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 two equations with two variables, which can then be solved using the same techniques.

Steps:

1. Solve one equation for one variable: Choose an equation and solve it for one of the variables in terms of the other two.

2. Substitute: Substitute the expression you found in step 1 into the other two equations. This will give you a system of two equations with two variables.

3. Solve the 2x2 system: Solve this smaller system using either elimination or substitution.

4. Substitute back: Substitute the values you found in step 3 back into the expression you found in step 1 to solve for the third variable.

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

The substitution method can be more complex than elimination for three-variable systems, and it's often preferable to use elimination. However, understanding substitution is still crucial for solving systems of equations in general.

Method 3: Using Matrices and Gaussian Elimination

For larger systems or when dealing with many systems, matrices and Gaussian elimination provide a more efficient approach. This involves representing the system as an augmented matrix and performing row operations to reduce it to row-echelon form or reduced row-echelon form.

The details of Gaussian elimination and matrix operations are beyond the scope of this introductory guide, but it's a powerful technique to learn for solving larger systems of linear equations efficiently.

Handling Special Cases

No Solution: If you arrive at a contradiction (e.g., 0 = 1), the system has no solution. Graphically, this means the planes do not intersect at a single point.

Infinitely Many Solutions: If you arrive at an identity (e.g., 0 = 0), the system has infinitely many solutions. Graphically, this means the planes intersect along a line or coincide.

Practical Applications

Solving systems of equations with three variables has numerous applications in various fields, including:

• Physics: Analyzing forces, motion, and circuits.
• Chemistry: Determining the composition of mixtures.
• Economics: Modeling supply and demand.
• Engineering: Solving structural problems and analyzing networks.
• Computer Graphics: Representing 3D objects and transformations.

Mastering the techniques for solving systems of three-variable equations provides a valuable skillset applicable to a wide range of quantitative problems. Remember to practice consistently, pay attention to detail, and choose the method that best suits the given system. With practice, you'll find that these seemingly complex problems become straightforward and efficient to solve.