Problem 42 - Algebra and Trigonometry

Question

What is a system of linear equations in three variables?

Step-by-Step Solution

Answer

A system of linear equations in three variables consists of three or more linear equations involving the same three variables. It can be represented in the form of \(a_1x + b_1y + c_1z = d_1\), \(a_2x + b_2y + c_2z = d_2\), \(a_3x + b_3y + c_3z = d_3\). The system may have a unique solution, infinite solutions, or no solution, and it can be solved using methods like substitution, elimination, or matrices.

Step 1: The Definition

A system of linear equations in three variables is a system that consists of three or more linear equations involving three variables. It can be represented as \[ \begin{cases} a_1x + b_1y + c_1z = d_1 \ a_2x + b_2y + c_2z = d_2 \ a_3x + b_3y + c_3z = d_3 \end{cases} \] where \(a_1\), \(b_1\), \(c_1\), \(d_1\), \(a_2\), \(b_2\), \(c_2\), \(d_2\), \(a_3\), \(b_3\), \(c_3\), \(d_3\) are constants and \(x\), \(y\), \(z\) are variables.

Step 2: Possibilities for Solutions

The system of linear equations can either have one unique solution, infinite many solutions, or no solution. \ 1. It has a unique solution if the three planes represented by the equations intersect at a single point. \ 2. It has infinite solutions if the three planes all coincide, meaning they are all the same plane. \ 3. It has no solution if the three planes do not intersect on a single line or at a single point.

Step 3: Solving the System

The system of linear equations in three variables can be solved through various methods such as substitution, elimination, or matrices. \ The goal is to simplify the system to a stage where it is easy to identify the solutions. The procedure involves eliminating one variable at a time until you are able to solve for one of the variables directly.