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Can a system of Linear Equations have exactly two distinct solutions

Linear systems of equations form the backbone of many mathematical and real-world applications. But how many solutions can such a system have? Broadly, there are only three possibilities for a system of linear equations:

  1. A unique solution: The system has one precise solution.
  2. Infinite solutions: The system represents overlapping equations or a family of solutions.
  3. No solution: The system represents contradictory equations that cannot coexist.

But what if someone claims that a system of linear equations can have exactly two distinct solutions? Let’s explore why this claim is not possible.

The Setup: Assuming Two Distinct Solutions

Suppose we have a linear system represented by \(\mathbf{A}\mathbf{x} = \mathbf{b}\), where \(\mathbf{A}\) is the coefficient matrix, \(\mathbf{x}\) is the vector of variables, and \(\mathbf{b}\) is the constant vector. Let’s assume this system has two distinct solutions, \(\mathbf{u}_1\) and \(\mathbf{u}_2\).

By definition of solutions:

\[ \mathbf{A}\mathbf{u}_1 = \mathbf{b} \quad \text{and} \quad \mathbf{A}\mathbf{u}_2 = \mathbf{b}. \]

Exploring the Claim

Now, let’s consider a new vector \(\mathbf{x}\) defined as a linear combination of \(\mathbf{u}_1\) and \(\mathbf{u}_2\):

\[ \mathbf{x} = \lambda \mathbf{u}_1 + (1 - \lambda)\mathbf{u}_2, \quad \text{where } \lambda \in \mathbb{R}. \]

To check if \(\mathbf{x}\) is also a solution, we substitute it into the system \(\mathbf{A}\mathbf{x} = \mathbf{b}\):

\[ \mathbf{A}\mathbf{x} = \mathbf{A} \big(\lambda \mathbf{u}_1 + (1 - \lambda)\mathbf{u}_2\big). \]

Using the linearity property of matrix multiplication:

\[ \mathbf{A}\mathbf{x} = \lambda \mathbf{A}\mathbf{u}_1 + (1 - \lambda)\mathbf{A}\mathbf{u}_2. \]

Substituting \(\mathbf{A}\mathbf{u}_1 = \mathbf{b}\) and \(\mathbf{A}\mathbf{u}_2 = \mathbf{b}\), we get:

\[ \mathbf{A}\mathbf{x} = \lambda \mathbf{b} + (1 - \lambda)\mathbf{b}. \]

Simplifying:

\[ \mathbf{A}\mathbf{x} = \mathbf{b}. \]

Thus, \(\mathbf{x}\) is also a solution to the system for any value of \(\lambda \in \mathbb{R}\).

The Contradiction

This result implies that if a system has two distinct solutions, it must have infinitely many solutions, as \(\lambda\) can take any real value. Hence, the assumption that a linear system can have exactly two distinct solutions leads to a contradiction.

Conclusion

A system of linear equations cannot have exactly two solutions. The nature of linear systems ensures that the solutions must fall into one of three categories: unique, infinite, or none. If two distinct solutions exist, the system inherently admits infinitely many solutions, as demonstrated by the linear combination argument.

The next time someone claims a linear system has exactly two solutions, you’ll know the mathematical reasoning to debunk it!