Consider the following system of equations. (a) The system has more than one solution when k = (b) The system has no solution when k =x+y+kz=6x+ky+z=9kx+y+z=7 Solution a)1/0/infinitely many b)0/1 x+3y-z=-4 4x-y+2z=3 2x-y-3z=1 the important concept here is linear dependence versus linear independence. As shown in the examples posted, linear dependence occurs when one equation in the system of equations can be shown to be a multiple of another. This is ultimately what Gaussian elimination or computing the determinant reveals. In this instance, there is no unique solution to the system of equations. Conversely, if the system of equations is linearly independent, then a unique solution does exist (though you still have to compute it, as is done in the examples in other answers). This can be visualized by graphing the equations, assuming a low order for the system. Linear dependence implies that two or more of the lines obtained when graphing the system are parallel or one on top of the other. Linear independence will result in a graph in which the various lines intersect at one point, that point being the solution to the system of equations..

1. Consider the following system of equations.
(a) The system has more than one solution when k =
(b) The system has no solution when k =x+y+kz=6x+ky+z=9kx+y+z=7
Solution
a)1/0/infinitely many
b)0/1
x+3y-z=-4
4x-y+2z=3
2x-y-3z=1
the important concept here is linear dependence versus linear independence. As shown in the
examples posted, linear dependence occurs when one equation in the system of equations can be
shown to be a multiple of another. This is ultimately what Gaussian elimination or computing the
determinant reveals. In this instance, there is no unique solution to the system of equations.
Conversely, if the system of equations is linearly independent, then a unique solution does exist
(though you still have to compute it, as is done in the examples in other answers).
This can be visualized by graphing the equations, assuming a low order for the system. Linear
dependence implies that two or more of the lines obtained when graphing the system are parallel
or one on top of the other. Linear independence will result in a graph in which the various lines
intersect at one point, that point being the solution to the system of equations.