Examples:

1. Find the unique values of variables a, b, and c.

   1a+2b+3c=18.1
2a-2b+4c=14.2
3a+5b-2c=6.9


   Equation two minus two times equation one yields $$-6b-2c=-22$$ While equation three minus 3 times equation one becomes $$-b-11c=-47.4$$ Next, we use these two new equations to solve for the variable c by eliminating b as shown below. $$-6b-2c+-6(-b-11c)=22+-6(-47.4)\underset{}{\to }64c=262.4\underset{}{\to }c=4.1$$ Plugging this value into either of the equations with variable a removed yields b=2.3. Plugging found values for c and b into any of the original 3 equations will cause an outcome of a=1.2.

2. Find the unique values of variables a, b, and c.

   3a+2b+3c=11.5
2a-2b-c=1.6
4a-2b-c=11


   Equation one plus equation two becomes $5a+2c=13.1$
   Equation one plus equation three becomes $7a+8c=22.5$

   Comment: Notice variable elimination order is not important.

   Using these two calculated equations one can solve for a by subtracting four times the first calculated equation from the second calculated equation. $$-13a=-29.9\underset{}{\to }a=2.3$$ Plugging the a value into either of the calculated equations gives $5(2.3)+2c=13.1\underset{}{\to }c=0.8$ . Finally plugging in numerical values of a and c into any of the original equations gives the result b=1.1.