Make up a system of three linear equations containing three...

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

Key Concepts

System of Linear Equations

A system of linear equations is a collection of equations involving the same set of variables. The goal is to find values for these variables that satisfy every equation simultaneously. Systems with three variables usually represent three-dimensional conditions where each equation corresponds to a plane in space.

Consistency and Inconsistency

A system is called consistent if there exists at least one set of values that satisfies all the equations, and inconsistent if no such set exists. Consistency is crucial in determining whether a system has a unique solution or infinitely many solutions.

Unique Solution

A unique solution occurs when all the equations in a system intersect at exactly one point, which represents the only set of values that satisfies all equations simultaneously. This usually means that the equations are independent and the system’s coefficient matrix is non-singular.

Infinitely Many Solutions (Dependent Systems)

Infinitely many solutions arise in dependent systems where the equations do not provide completely independent constraints. Instead, one or more of the equations can be expressed as a linear combination of the others, meaning they represent the same geometric object, leading to an overlap of solution sets.

No Solution (Inconsistent Systems)

A system has no solution when its equations represent parallel geometric objects, such as planes that never intersect. In this case, the conditions set by the equations are contradictory, making it impossible to find any combination of variable values that satisfies all the equations at once.

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