Problem 73 - Introductory Algebra for College Students

Question

When using the addition method, how can you tell if a system of linear equations has infinitely many solutions?

Step-by-Step Solution

Answer

Upon using the addition method, if you end up with an identity like \(0 = 0\), it indicates that the system of linear equations has infinitely many solutions.

Step 1: Understand Addition Method

The addition method involves adding corresponding terms of the equations in the system. If the system is consistent and the equations are dependent, it will result in a statement that is always true, such as \(0 = 0\).

Step 2: Identify Infinitely Many Solutions

A system has infinitely many solutions when, upon using the addition method, you end up with an identity, a true statement. This is typically represented as \(0 = 0\) or something similar.

Step 3: Example Case

Consider a system of two equations: \(2x + 3y = 6\) and \(4x + 6y = 12\). If the equations are added together, the result is \(6x + 9y = 18\). This equation is proportional to both original equations, essentially repeating information. Thus, no new, unique solution is provided. This indicates that the system has infinitely many solutions.