Let there be a set of $3$ equations as follows :
\begin{align*}
a_1x + b_1 y + c_1 z &= d_1 \\
a_2x + b_2 y + c_2 z &= d_2 \\
a_3x + b_3 y + c_3 z &= d_3
\end{align*}
Do we have a set of simple rules or conditions under which the above set of linear equations will have a unique solution, no solution or infinite solutions? I tried to find the information on this on the web but couldn't find any.
I have read about this back in my college and I know that we can determine the uniqueness of solution based on the relationship between the rank of the coefficient matrix, the rank of an augmented matrix and the number of variables in the equation but I am looking for some simple ratio relationships between the coefficients of variables as we have for a linear system of equations in 2 variables. Please help !!!
Best Answer
Simplest way:
These equations are also Eq. of planes in 3D.
If any two panes are parallel then no solution.
If two of them are coincident.Then planes meet in a line and there are many solutions.
If all three planes are coincident, then many solutions.
Else, let $z=k$ and solve first two equations get $x,y$ in terms of $k$, put them in third equation. Three mutully exclusive things can happen.
1-$k$ gets determined, so unique solution: Planes meeting in a point.
2-$k$ disappears leaving a true statement like $3=3$, so many solutions possible for any real value of $k$: Planes meeting in a line.
3-$k$ disappears leaving a flase statement like $3=4$, so no solutions possible for any real value of $k$:Planes forming an open prism.
These situation can also be told in terms of Cramer determinants, adjoint or rank. of a matrix. When no solutions, system is called inconsistent. If unique or many solutions, the system is called consistent.