## Summary: Possibilities for the Solution Set of a System of Linear Equations

## Problem 288

In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems.

Determine all possibilities for the solution set of the system of linear equations described below.

**(a)** A homogeneous system of $3$ equations in $5$ unknowns.

**(b)** A homogeneous system of $5$ equations in $4$ unknowns.

**(c)** A system of $5$ equations in $4$ unknowns.

**(d)** A system of $2$ equations in $3$ unknowns that has $x_1=1, x_2=-5, x_3=0$ as a solution.

**(e)** A homogeneous system of $4$ equations in $4$ unknowns.

**(f)** A homogeneous system of $3$ equations in $4$ unknowns.

**(g)** A homogeneous system that has $x_1=3, x_2=-2, x_3=1$ as a solution.

**(h)** A homogeneous system of $5$ equations in $3$ unknowns and the rank of the system is $3$.

**(i)** A system of $3$ equations in $2$ unknowns and the rank of the system is $2$.

**(j)** A homogeneous system of $4$ equations in $3$ unknowns and the rank of the system is $2$.

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