Problem 1 - College Algebra with Corequisite Support

Question

Can a linear system of three equations have exactly two solutions? Explain why or why not.

Step-by-Step Solution

Answer

No, a system of three linear equations cannot have exactly two solutions.

Step 1: Understanding the Problem

We are asked whether a linear system of three equations can have exactly two solutions. In general, a solution to a system of linear equations is a set of values for the variables that makes all equations true simultaneously.

Step 2: Analyzing Possibilities for Solutions

A linear system can have no solution, exactly one solution, or infinitely many solutions. A system cannot have exactly two solutions because linear equations represent planes in a three-dimensional space, which can intersect in a point (one solution), a line (infinitely many solutions), or not at all (no solutions).

Step 3: Studying Intersection of Planes

In three dimensions, the intersection of three planes is crucial. They can meet at a single point (one solution), be parallel (no solutions), or intersect in such a way that they form a line (infinitely many solutions). However, three planes cannot intersect in exactly two distinct sets of solutions.

Step 4: Drawing a Conclusion

Given the behavior of planes, it's clear that each system follows specific geometric interpretations in space, confirming that a system of three linear equations cannot have exactly two solutions.

Key Concepts

Solutions of Linear Systems

A system of linear equations consists of several equations with the same variables, and solving it means finding values for the variables that satisfy all the equations simultaneously. In simpler terms, we are looking for a common solution that works for all the equations in the system at once.

There are three primary scenarios for the number of solutions in any linear system:

• **No Solutions:** This occurs when the lines or planes do not intersect at any point. For example, this could happen if the planes are parallel.
• **One Solution:** This occurs when the lines or planes meet at a single unique point. In 3D, this means the three planes intersect at exactly one point.
• **Infinitely Many Solutions:** This happens when the lines or planes coincide or form a single line of intersection. This means there is not just one set of values that solve the system, but an entire set of values along a line.

The question of two exact solutions doesn’t fit into these scenarios because our geometric understanding indicates that such a situation can't exist given the nature of linear systems.

Intersection of Planes

In three-dimensional space, each linear equation represents a plane. Understanding how these planes intersect helps in visualizing and solving the system.

When considering the intersection of three planes, here’s what we need to know:

• **Single Point Intersection:** This occurs when all three planes meet precisely at one spot. It represents the one unique solution to the system. This is quite like the tip of a pyramid where all the sides converge.
• **No Intersection (Parallel Planes):** If at least two of those planes run parallel to each other, they may never meet, resulting in no solution.
• **Line of Intersection:** It's possible for three planes to intersect in a line. This is a scenario for infinitely many solutions because any point on that line is a common solution to all three equations.

Understanding these intersections is essential because it demonstrates why there can't be exactly two solutions from three planes. The geometry just doesn’t allow for it.

Geometric Interpretation of Solutions

A geometric interpretation helps us visualize and understand possible solutions to a linear system. Imagine the plane being a sheet of paper in space. The intersection patterns explain the solutions:

• **Point of Intersection:** Each plane is like a flat surface in space, so when they meet at a single point, that's akin to three papers touching just one corner together. This represents a single solution.
• **Parallel Planes and No Common Point:** Think of railroad tracks that run parallel. They never touch, which parallels the idea of no solution.
• **Intersecting at a Line:** Consider a book with pages that all hinge at a central spine. When planes intersect along a line, it's like each page touches the spine at many points. This signifies infinitely many solutions, akin to swinging the book open to any angle along the spine.

This visualization aids in understanding why a system can't have exactly two solutions. Graphically, no plausible arrangement of the planes allows for exactly two distinct intersection points, reinforcing the theory behind the nature of linear systems.