Thread: linear equations

why cant a set of two linear equations in two unkowns have exactly two solutions.While it is easy to see geometrically. How can we prove it algebrically ?

using simultaneous equation..

Originally Posted by E(i)lusiveReality

why cant a set of two linear equations in two unkowns have exactly two solutions.While it is easy to see geometrically. How can we prove it algebrically ?

If the equations are homogeneous, the solutions form a vector space. The only finite vector space is the zero vector space.

If the equations are not homogeneous, a general solution is any single solution plus solutions of the associated homogeneous equations. Thus there is either no solution, a unique solution or infinitely many solutions.

This also applies to n equations and m unknowns.

If you think about it geometrically, it is easy to understand. Each equation is a line in the plane and the solution is the (x,y) coordinate where the lines cross. If they are parallel, there is no solution.