While working on a problem, I came upon a system of four equations. Namely, $$\begin{align*} & 4+b=d\\ & 4c=4d+36+i\\ & 2c=d+2i+6\\ & 2c=2d+2b+3\end{align*}\tag1$$ And I'm wondering if there are other conventional methods for solving system of $4$ equations without the use of Mathematica, or matrices since I won't be able to have a calculator on the test.
The method that I proposed was to rearrange all the variables to the LHS to get$$\begin{align*} & b-d=-4\\ & 2c-d-2i=6\\ & -2b+2c-2d=3\\ & 4c-4d-i=36\end{align*}\tag{2}$$ And multiply the first three equations by $A,B,C$ respectively and add them together. Thus, getting$$(A-2C)b+(2B+2C+4)c+(-A-B-4)d+(-2B-1)i=-4A+6B+3C+36\tag{3}$$ And to solve for $b$, set the other coefficients equal to zero to find $A,B,C$ and thus, $b$ is found by$$b=\frac{-4A+6B+3C+36}{A-2C}\tag{4}$$
The only problem with that method is how lengthy it gets. You can't reuse $A,B,C$; the values change for each variable you want to find. So my actual questions...
Questions:
- How would you solve this system of $4$ equations?
- Can this method be generalized to any system of $4$ equations?
- Is there an underlying method to solve an $n$ system of equations? (Miscellaneous question; you don't have to answer that!)