Question

Make up a system of three linear equations containing three variables that have: (a) No solution (b) Exactly one solution (c) Infinitely many solutions Give the three systems to a friend to solve and critique.

Short answer

System 1: No solutionSystem 2: Exactly one solutionSystem 3: Infinitely many solutions

Step-by-step solution

Understanding the Requirements

Determine the type of linear equation systems based on the given conditions: (a) no solution, (b) exactly one solution, and (c) infinitely many solutions.

Formulating a System with No Solution

Set up three linear equations that represent parallel planes, which do not intersect. For example: \[ \begin{cases} x + y + z = 1 \ x + y + z = 2 \ x + y + z = 3 \ \text{These equations represent parallel planes that never meet, thus there is no solution.} \ \text{System 1:} \ x + y + z = 1 \ x + y + z = 2 \ x + y + z = 3 \ \right. \]

Formulating a System with Exactly One Solution

Set up three linear equations that intersect at a single point. For example: \[ \begin{cases} x + y + z = 6 \ 2x - y + 3z = 14 \ -x + y + 2z = 8 \ \text{These equations represent planes that intersect at exactly one point.} \ \text{System 2:} \ x + y + z = 6 \ 2x - y + 3z = 14 \ -x + y + 2z = 8 \ \right. \]

Formulating a System with Infinitely Many Solutions

Set up three linear equations that represent the same plane or intersect in a line. For example: \[ \begin{cases} x + y + z = 5 \ 2x + 2y + 2z = 10 \ 3x + 3y + 3z = 15 \ \text{These equations represent the same plane, thus having infinitely many solutions.} \ \text{System 3:} \ x + y + z = 5 \ 2x + 2y + 2z = 10 \ 3x + 3y + 3z = 15 \ \right. \]

Presenting the Systems

Give these three systems of equations to a friend to solve and critique. Ensure they understand the types of solutions they need to identify.

Key concepts

No Solution

In some systems of equations, it is impossible to find a set of values that satisfy all the equations at the same time. This situation is called 'no solution'. For example, let's take the system of linear equations:

• \(x + y + z = 1\)
• \(x + y + z = 2\)
• \(x + y + z = 3\)

These equations represent three parallel planes that never intersect. Because these planes do not share any common points, it's impossible to find a single set of \(x\), \(y\), and \(z\) values that satisfy all the equations.
This means we have no solution in this system.
When you encounter three planes like this that never meet, they don't have any intersection point in common. It's essential to identify such systems because searching for a common solution would be futile!

Unique Solution

A system of linear equations can also have exactly one solution. This is known as a 'unique solution', where the equations intersect at a single point. Consider the following system:

• \(x + y + z = 6\)
• \(2x - y + 3z = 14\)
• \(-x + y + 2z = 8\)

In this system, the equations represent three planes that intersect at exactly one point. This single point, where all the planes meet, is the unique solution to the system.
In a practical sense, it means that there is one and only one set of values for \(x\), \(y\), and \(z\) that satisfy all three equations. In terms of geometry, we find that the intersection of the planes is precise and not broad. This is important to recognize, as it implies a deterministic result.

Infinitely Many Solutions

Sometimes, a system of equations can have infinitely many solutions. This usually happens when the equations represent the same plane or intersect along a line. For instance, consider the following system:

• \(x + y + z = 5\)
• \(2x + 2y + 2z = 10\)
• \(3x + 3y + 3z = 15\)

In this case, all three equations are multiples of each other. They represent the same plane in space. Since the equations overlap completely, every point on this plane is a solution. Therefore, there are infinitely many solutions.
Another instance could be when three planes intersect along a line, whereby the line of intersection would provide infinite points common to all planes. Necessary to realize this scenario is to understand that the system does not have a finite or single solution but rather an unlimited set of solutions, reflecting continuity and dependency among the equations.