find all the values of a and b so that the system has a) no solution b) 1 solution c) exactly 3 solutions and 4) infinitely many solutions
Apply Gaussian Elimination to the reduced matrix. $$ \pmatrix{1 & -1 & 2 & 4\\3 & -2 & 9 & 14\\2 & -4 & a & b} \to \pmatrix{1 & -1 & 2 & 4\\0 & 1 & 3 & 2\\0 & -2 & a-4 & b-8} \to \pmatrix{1 & -1 & 2 & 4\\0 & 1 & 3 & 2\\0 & 0 & a+2 & b+4} $$ So
- if you want a unique solution, $\frac{b+4}{a+2}$ must be defined, i.e. $a \neq -2$.
- if you want infinite number of solutions, $a=-2$ and $b = -4$ removes the last equation.
- if you want no solutions, $a=-2$ and $b \neq -4$.