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Math
Emeritus
Thanks
PF Gold
P: 38,898

## A system of equations with more unknowns than equations

"Consistent" means that there exist at least one solution. "Inconsistent" means there is no solution.

Examples with 3 unknowns and 2 equations are:
x+ y+ z= 2, x+ y- z= 4: consistent

and x+ y+ z= 2, x+ y+ z= 1: inconsistent

With x+ y+ z= 2 and x+ y- z= 4, I can add the equations to eliminate z and get 2x+ 2y= 6 so that x+ y= 3. I can choose any value for, say, x, then solve for y, then z. For example, if x= 1, then y= 3- 1= 2 so that the first equation becomes 1+ 2+ z= 3+ z= 2 and z= -1. x= 1, y= 2, z= -1 is a solution. But I could as well, take x= 2 and get y= 3- 2= 1, 3- z= 4, x= -1. In fact, (x, 3- x, -1) is a solution for x any real number.

With x+ y+ z= 2, x+ y+ z= 1, subtracting the second equation from the first, we get 0= 1 which is impossible. If fact the two left sides are identical. If x+ y+ z= 2, then it can't possibly be equal to 1 whatever x and y are!