Problem 55 - Introductory Algebra for College Students

Question

What is a solution of a system of linear equations?

Step-by-Step Solution

Answer

A solution of a system of linear equations is a set of variable values that satisfy all the equations in that system simultaneously.

Step 1: Understanding Linear System

A system of linear equations is a collection of one or more linear equations involving the same variables. For instance, a system of two equations in two variables looks like this: \[ ax + by = m \] and \[ cx + dy = n \]

Step 2: Defining Solution for Linear System

A 'solution' of this system is a pair (x, y) of real numbers that makes both equations true when these values are substituted into the equations. In other words, (x, y) is a solution if the left-hand side equals the right-hand side for both equations when x and y are replaced by the solutions.

Step 3: Methods to Find Solutions

There are numerous methods we can use to find the solution to a system of linear equations: Substitution method, Elimination method, and Matrix method. Each of these methods involves procedures that require algebraic manipulations.