is it possible when the number of unknowns is less than the linear equations to have a unique solution.

Question

X₁₊2X₂=0

2X₁+X₃=0

-X₁+2X₃=0

4X₁+7X₃=0

Answer 1

last 3 linear equations have just 2 unknowns, x1 and x3

if 2 of the 3 had been linearly dependent, multiples of each other, then the system of 3 equations, 2 unknowns would have a unique solution

but the given last 3 equations have no solution. DNE, a real solution Does Not Exist, as they are contradictory. Graph the 3 equations and they do not intersect in one point.

as written the problem has 4 equations 3 unknowns, x1, x2 & x3, less unknowns than the number of equations

but also contradictory with no solution. But in general, it's possible for 4 equations 3 unknowns to have a unique solution