Solving a System of 3 Linear Equations with 4 Variables: A Comprehensive Guide

Introduction to Linear Equations

Linear equations are a fundamental part of algebra and play a significant role in various fields, including mathematics, engineering, and physics. When dealing with a system of linear equations, the problem can often become intriguing when the number of unknowns (variables) exceeds the number of equations, leading to an underdetermined system. This article focuses on a specific case where we have three linear equations with four variables. Let's explore the steps to solve such a system and understand the concept of underdetermined systems.

Understanding the Basics

When dealing with a system of linear equations, it's crucial to grasp the term underdetermined system. An underdetermined system occurs when the number of unknowns (variables) exceeds the number of equations. For instance, if we have three equations with four unknowns, the system is said to be underdetermined, and it typically leads to an infinite number of solutions. The goal is to express some variables in terms of the free variables to describe the general solution.

Steps to Solve the System

Step 1: Write the Equations

Let's begin with a general system of three linear equations with four variables:

a1x b1y c1z d1w e1

a2x b2y c2z d2w e2

a3x b3y c3z d3w e3

Step 2: Set Up the Augmented Matrix

Representing the system using an augmented matrix can simplify the process:

[a1 b1 c1 d1 e1
a2 b2 c2 d2 e2
a3 b3 c3 d3 e3]

Step 3: Row Reduction

The next step is to perform row reduction to transform the matrix into row echelon form (REF) or reduced row echelon form (RREF). This process involves a series of row operations such as swapping rows, multiplying a row by a constant, and adding multiples of one row to another. Gaussian elimination is a method commonly used to achieve this.

Step 4: Identify Free Variables

Once the matrix is in REF or RREF, it becomes apparent that at least one variable is free, as the system is underdetermined. This means that some variables (usually the ones with no pivot in the corresponding column) can be expressed in terms of the free variables. These free variables can take any value, leading to an infinite number of solutions.

Step 5: Express Solutions

After converting the matrix into REF or RREF, the equations should look similar to:

x p1y q1z r1w s1

y t1z u1w v1

z v1w u1

Here, p1, q1, r1, s1, t1, u1, v1, and u1 are constants derived from the row reduction process.

Step 6: Write the General Solution

To express the solution set, you can write the variables in terms of the free variables. For instance, if w is a free variable, you might write:

x s1 - p1y - q1z - r1w

y v1 - t1z - u1w

z u1 - v1w

w w

where w can take any value, leading to an infinite number of solutions.

Example

Consider the following system:

x 2y z w 1

2x y 3z w 2

3x 4y z 2w 3

Arrange the equations into an augmented matrix and perform row reduction:

[1 2 1 1 1
2 1 3 1 2
3 4 1 2 3]

Create the RREF:

[1 0 0 0 0
0 1 0 0 0
0 0 1 1 1]

From the RREF, we see that w is a free variable, so we express x, y, and z in terms of w:

x 0 - 0y - 0z - 0w 0

y 0 - 0y - 0z - 0w 0

z 1 - 1w 1 - w

w w

Thus, the general solution is:

x 0

y 0

z 1 - w

w w

Conclusion

In conclusion, solving a system of three linear equations with four variables involves expressing some variables in terms of free variables and writing the general solution. The specific steps can vary based on the coefficients and constants in your equations. Understanding the process and the concept of underdetermined systems can help in solving similar problems more efficiently.