Problem 59 - Introductory Algebra for College Students

Question

Explain how a linear system can have infinitely many solutions.

Step-by-Step Solution

Answer

A linear system has infinitely many solutions if the equations are dependent, implying that they represent the same line or plane. This means that there is a whole set of values for the variables that will satisfy all the equations.

Step 1: Understanding Infinitely Many Solutions

In a system of linear equations, having 'infinitely many solutions' means that there is not a unique solution for the variables. Instead, any value can be substituted for one variable, and the other variables would adjust accordingly to satisfy the equations. This typically happens when the equations in the system are dependent, meaning one equation is a multiple of another.

Step 2: Analyzing Linear Systems

In a three-variable system, for example, equations are represented in a 3-dimensional space. When three planes intersect at a single line, there will be infinitely many solutions. Similarly, for a two-variable system, when the equations represent the same line, there are infinitely many points of intersection along the entire line, hence infinitely many solutions.

Step 3: Representing Infinitely Many Solutions algebraically

In terms of algebra, consider this system: \[ E1: ax + by = c \] and \[ E2: dx + ey = f \]. If \[ E2 \] is a multiple of \[ E1 \] that is, \( d = ka, e = kb, f = kc \) for some constant \( k \), then any solution for \( E1 \) is a solution for \( E2 \). Hence, there exist an infinite set of solutions.