# The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns

## Problem 295

Determine all possibilities for the number of solutions of each of the system of linear equations described below.

(a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution.

(b) A homogeneous system of $5$ equations in $4$ unknowns and the rank of the system is $4$.

(The Ohio State University, Linear Algebra Midterm Exam Problem)

## Hint.

See the post Summary: possibilities for the solution set of a system of linear equations to review how to determine the number of solutions of a system of linear equations.

## Solution.

Let $m$ be the number of equations and $n$ be the number of unknowns in the given system.
Note that the information $m > n$ tells us nothing about the possibilities for the number of solutions.

### (a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution.

The system has at least one solution $x_1=0, x_2=-3, x_3=1$, hence it is consistent.
Thus, the system has either a unique solution or infinitely many solutions. Since $m > n$, we cannot narrow down the possibilities further. Thus, the possibilities are either a unique solution (which must be $x_1=0, x_2=-3, x_3=1$) or infinitely many solution.

### (b) A homogeneous system of $5$ equations in $4$ unknowns and the rank of the system is $4$.

Since the system is homogeneous, it has always the zero solution, hence it is consistent.
The fact $m > n$ gives no new information. But since the rank $r$ of the system is equal to the number of unknowns $n$, there is no free variable. ($n-r=0$.) Thus, the system must have a unique solution, which is the zero solution.

## Comment.

This is one of the midterm exam 1 problems of linear algebra (Math 2568) at the Ohio State University.

Some students wrongly concluded from the condition $m >n$. Again, note that the condition $m > n$ gives no new information at all.

## Midterm 1 problems and solutions

The following list is the problems and solutions/proofs of midterm exam 1 of linear algebra at the Ohio State University in Spring 2017.

1. Problem 1 and its solution (The current page): Possibilities for the solution set of a system of linear equations
2. Problem 2 and its solution: The vector form of the general solution of a system
3. Problem 3 and its solution: Matrix operations (transpose and inverse matrices)
4. Problem 4 and its solution: Linear combination
5. Problem 5 and its solution: Inverse matrix
6. Problem 6 and its solution: Nonsingular matrix satisfying a relation
7. Problem 7 and its solution: Solve a system by the inverse matrix
8. Problem 8 and its solution:A proof problem about nonsingular matrix

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