Problem 1 - Precalculus: Mathematics for Calculus

Question

If a system of linear equations has infinitely many solutions, then the system is called __________. If a system of linear equations has no solution, then the system is called _________.

Step-by-Step Solution

Answer

The system is called 'dependent' with infinitely many solutions and 'inconsistent' with no solution.

Step 1: Understanding Consistent Systems

A system of linear equations is referred to as consistent if it has at least one solution. If the system has infinitely many solutions, it is specifically termed as a 'dependent' system because the equations are dependent on each other, sharing multiple solutions.

Step 2: Recognizing Inconsistent Systems

An inconsistent system is a set of equations that has no solutions. This typically occurs when the equations represent parallel lines that never intersect or the constraints of the system are contradictory, making it impossible for any set of variables to satisfy all equations simultaneously.

Key Concepts

Consistent Systems

Understanding consistent systems is crucial when dealing with linear equations. These systems are characterized by having at least one solution. This means that there is at least one set of values for the variables involved that satisfies all the equations in the system. For instance, when graphed, the equations representing a consistent system might intersect at a single point or lie on top of each other, sharing all their infinite points of intersection.
There are two types of consistent systems:

• Independent: This is where the system has exactly one solution, usually represented graphically as two lines crossing at a single point.
• Dependent: This occurs when the system has infinitely many solutions, meaning all equations essentially describe the same line or plane.

Inconsistent Systems

An inconsistent system of linear equations is a scenario where there are no solutions. Practically, this means there's no set of values for the variables that can satisfy all the equations simultaneously. Graphically, this is often seen as parallel lines, which, as they never meet, signify no point of intersection.
Inconsistency often arises when the equations contradict each other. For example, you might have one equation demanding one thing and another insisting on something else that cannot coexist—like saying a point must lie simultaneously on two parallel lines. This makes it impossible for any point to satisfy both equations. Thus, the set of linear equations has zero solutions.

Dependent Systems

Dependent systems are a subset of consistent systems. They are characterized by having infinitely many solutions. This occurs when all the equations in the system, though they may look different at first, represent the same line or plane. Thus, every solution for one equation is also a solution for the others.
When graphed, a dependent system's equations overlap completely. It means they share all the same points or values. For example, if you have an equation that simplifies into another within the system, you're working with a dependent system. In practical terms, this can happen when equations are simply multiples or variations of each other, leading to a redundancy in information that allows for infinitely many solutions.