Problem 36 - College Algebra

Question

What is a system of linear equations in three variables?

Step-by-Step Solution

Answer

A system of linear equations in three variables is a set of equations that use the same three variables, normally x, y, and z. They are first degree equations solved simultaneously with the solutions representing an intersection point on a three-dimensional graph.

Step 1: Definition of a 'System of Equations'

A 'system of equations' is a set or collection of equations that are solved simultaneously. The solution of a system of equations is an ordered pair (or triplet or more for more variables) that makes all equations in the system true.

Step 2: Understanding 'Linear Equations'

Linear equations are equations of the first degree, meaning that the variables are only to the power of 1 and not subjected to functions like square roots or exponential functions. They are presented in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are real numbers.

Step 3: The 'Three Variables'

The 'three variables' in this context are typically represented as \(x\), \(y\), and \(z\). Each variable will correspond to a dimension in a three-dimensional graph, and the system of equations will intersect at a single point in this graph.

Step 4: Concept Integration

So, a system of linear equations in three variables is a collection of linear equations that all share the same variables \(x\), \(y\), and \(z\). The system is solved by finding the values of the variables that satisfy all the equations simultaneously.