Problem 42 - College Algebra

Question

Writing in Mathematics 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 linear equations in the variables \(x\), \(y\), and \(z\) in the form \(ax + by + cz = d\), \(ex + fy + gz = h\), and \(ix + jy + kz = l\). The variables \(x\), \(y\), and \(z\) can be any real numbers that satisfy all the equations in the system. The system might have a unique solution, infinite solutions, or no solution.

Step 1: Defining the General Form

A system of linear equations in three variables \(x\), \(y\), and \(z\) is a collection of linear equations that may be written in the form \(ax + by + cz = d\), \(ex + fy + gz = h\), and \(ix + jy + kz = l\), where \(a, b, c, d, e, f, g, h, i, j, k, l\) are constants.

Step 2: Explanation of Variables

The variables \(x\), \(y\), and \(z\) can be any real number that satisfies all of the equations in the system. This means that if you substitute the variables by their corresponding values in the equations, the resulting expressions would be true.

Step 3: Elaborating on Possible Solutions

A system of linear equations in three variables might have a unique solution (a single triple of real numbers), infinite solutions (infinitely many triples of real numbers), or no solution (no triple of real numbers that satisfies all equations).

Key Concepts

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It's a foundational topic in applied mathematics, engineering, and computer science. The concept of linear equations is at the heart of this field, defining relationships between algebraic variables where each term is either a constant or the product of a constant and a single variable. When these relationships are graphed in two dimensions, they appear as straight lines; in three dimensions, they can form planes or lines.

In the context of our exercise, we consider linear equations with three variables \(x, y, z\). These equations can represent geometric planes in a three-dimensional space. When we work with a system of such equations, we're essentially looking for points where these planes intersect. Depending on the nature of these planes – parallel, identical, or intersecting at a point or a line – we have different types of solutions. Understanding the geometric interpretations can help to visualize the problem and predict the nature of its solutions.

Solving Linear Systems

Solving a system of linear equations involves finding the values of the variables that satisfy all the equations simultaneously. There are several methods to tackle these systems, with no single 'best' approach. Popular methods include substitution, where we solve one equation for one variable and substitute this value into the others; elimination, which involves adding or subtracting equations to eliminate one variable and reduce the system to two equations; and matrix methods like row reduction or using determinants.

In the case of a system of three variables, we're working in a three-dimensional space and the goal is to find where three planes meet. This could lead to a single point of intersection (a unique solution), a line of intersection (infinite solutions), or no point of intersection at all (no solution). Matrix methods can be particularly powerful because they are systematic and can be applied easily by computer algorithms, which is an essential aspect in real-world applications where systems can have hundreds or even thousands of variables.

Algebraic Variables

Algebraic variables are symbols used to represent unknown quantities, usually denoted by letters such as \(x, y\), and \(z\). In a system of linear equations, these variables are interconnected through linear relationships. The essence of solving such a system is to pinpoint the exact numerical values for each variable that will satisfy all presented equations.

For a system involving three variables, we're looking for a set of three values, one for each variable, that holds true across all equations. This trio of numbers is often referred to as a solution 'triple'. The importance of algebraic variables goes beyond mere placeholders for numbers; they enable us to describe and solve problems abstractly without needing specific numerical values. This abstraction is crucial, as it allows for the application of solutions to a wide range of problems with similar structures.