Equations with three variables: Let the equations be $$ a_{1}x+b_{1}y+c_{1}z$$=$$d_{1}$$, $$ a_{2}x+b_{2}y+c_{2}z$$=$$d_{2}$$ and $$ a_{3}x+b_{3}y+c_{3}z$$=$$d_{3}$$. Here we define the following matrices: $$$D=\begin{bmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3}\end{bmatrix}$$$, $$$D_{x}=\begin{bmatrix} d_{1} & b_{1} & c_{1}\\ d_{2} & b_{2} & c_{2}\\ d_{3} & b_{3} & c_{3}\end{bmatrix}$$$, $$$D_{y}=\begin{bmatrix} a_{1} & d_{1} & c_{1}\\ a_{2} & d_{2} & c_{2}\\ a_{3} & d_{3} & c_{3}\end{bmatrix}$$$, $$$D_{z}=\begin{bmatrix} a_{1} & b_{1} & d_{1}\\ a_{2} & b_{2} & d_{2}\\ a_{3} & b_{3} & d_{3}\end{bmatrix}$$$
- Find the determinant of $$D$$ = $$a_{1}(b_{2}c_{3}-c_{2}b_{3})$$ + $$b_{1}(c_{2}a_{3}-a_{2}c_{3})$$ + $$c_{1}(a_{2}b_{3}$$-$$b_{2}a_{3})$$
- Similarly find the determinant of Dx, Dy and Dz
- If $$ \textrm{determinant of D}\neq 0$$, there exists a unique solution
- If $$ \textrm{determinant of D} = 0$$ and at least one of the determinants of Dx, Dy or Dz are non-zero then the system of equations has no solution
- If $$ \textrm{determinant of D} = 0$$ and all three determinants Dx, Dy and Dz are zero then there are infinitely many solutions
