Set Of Linear System

Question

How will you represent the solution set of a linear system in terms of set of points if the system has infinite solutions? For example write the solution of the linear system: $$3x+5y-7z=9 \\ x-6y+3z=1$$

Answer 1

If you need basic solution then the answer is this: Consider x=0, then the system of equation is , $5y-7z=9$ , $-6y+3y=1$ The solution of these two equations are $y=\frac{-59}{27} $ and $z=\frac{-34}{27}$ . Therefore the one basic solutions is $(0,\frac{-59}{27},\frac{-34}{27})$. Consider y=0 , then the system of equation is $3x-7z=9$ and $x+3y=1$. The solution is $x=\frac{17}{8}$ and $z=\frac{-3}{8}$.Therefore one of the other basic solution is $(\frac{17}{8},0,\frac{-3}{8})$. Now finally consider z=0, then the system of equation is , $3x+5y=9$ ,$x-6y=1$. The solution is $x=\frac{59}{23}$ and $y=\frac{6}{23}$ . Then the final basic solution is $(\frac{59}{23},\frac{6}{23},0)$ So the basic solutions are$ (0,\frac{-59}{27},\frac{-34}{27})$,$$(\frac{17}{8},0,\frac{-3}{8}),(\frac{59}{23},\frac{6}{23},0)$$.
These solutions are called optimal basic solution. If you want all solutions(not the basic) you have to substitute any variable from these two equations. Then you find one variable from this two's. Then take congruence.I substitute $x$ and found $23y-16y=6$. Taking congruence we get, $23y\equiv 6\pmod {16}$ . There you can get your all answers.