Problem 63 - Precalculus with Limits

Question

Writing Briefly explain whether it is possible for a consistent system of linear equations to have exactly two solutions.

Step-by-Step Solution

Answer

No, it is not possible for a consistent system of linear equations to have exactly two solutions.

Step 1: Explain

No, it is not possible. A consistent system of linear equations can have either exactly one solution (independent) or infinitely many solutions (dependent).

Step 2: Reasoning

If a system has two distinct solutions, then any linear combination of those solutions is also a solution (by linearity). This means there must be infinitely many solutions. Therefore, exactly two solutions is impossible.

Key Concepts

Consistent System

In the context of linear algebra, a consistent system is one where there exists at least one set of values for the variables that satisfies all equations simultaneously. Put simply, in a consistent system, the equations 'agree' with each other on at least one answer.

Imagine drawing several lines on a graph, each representing an equation in two-dimensional space; a consistent system is where these lines intersect at least once. This intersection represents a solution to the system. While consistent systems can either have a single unique solution or infinitely many solutions, it’s important to note that having exactly two solutions is not possible with linear equations since their intersection will always be a single point (or a line of intersection in the case of infinitely many solutions).

Unique Solution

When a consistent system has a unique solution, it means that all the lines, planes, or hyperplanes intersect at one and only one point. In mathematical terms, it is the solitary solution to the system. This scenario occurs when the equations are independent of each other, meaning no equation is a multiple or combination of the others.

Graphical Interpretation

For two variables, think of two non-parallel lines crossing at a point on a graph. This point of intersection is the unique solution, where both the x and y values satisfy both equations. The concept extends to higher dimensions too, where, for example, three planes might intersect at a single point in three-dimensional space.

Infinitely Many Solutions

A system of linear equations may have infinitely many solutions if all the equations represent the same line, plane, or hyperplane. This happens when each equation is a multiple of the others or, in other words, they are dependent.

How to Visualize Infinitely Many Solutions

For a simple two-variable case, picture a line on a graph. Any set of points that fall on this line is a solution to the equation of the line. Now, if a second equation represents the exact same line, every point on the line satisfies both equations, hence, there are infinitely many solutions. In higher dimensions, it's analogous to multiple planes lying exactly on top of each other.

No Solution

On the flip side of consistency is a system that has no solution—referred to as an inconsistent system. This occurs when there's no possible value or set of values for the variables that would satisfy all the equations simultaneously. Graphically, this means that the lines or planes do not intersect at any point.

An Example with Parallel Lines

Consider two parallel lines, which by definition, never meet. If these lines represent two equations in a two-variable system, there's no single point that lies on both lines. An extension of this concept to higher dimensions: two or more planes that are parallel won't intersect, similarly yielding no solution.