When presented with $3$ equations, to solve for $a$, $b$ and $c$ for me becomes a guessing game. If I have just two equations for example $3a + 2b = 5$ and $6a – 7b = 8$, you could easily solve for $a$ and $b$ through elimination or substitution. However, when I get three equations, e.g.,
$$\begin{aligned} 4a + 2b + c &= -3 \\ a + b + c &= -1 \\ a + b + c &= -1 \end{aligned}$$
it becomes a guessing game.
I usually start with elimination for a system of three equations but my biggest problem is substituting or elimination to get the same equation hence going around in circles. I then do blind/random substitutions to try and eliminate one variable, but I have no idea what I am doing or if it will work although I do get there in the end. Is this normal? Is there a set of steps or method which will be able to solve 3 for three unknowns without guessing randomly?
Best Answer
As Will Jagy comments, Gaussian Elimination is the way to go, especially to account for situations where there are no or infinitely many solutions.
In the scenario where there is a unique solution, it can be useful to think geometrically. The solutions of a linear equation in three variables form a plans in 3D space, and as long as two equations do not define parallel planes, the solution of two equations here is a line.
What we can do is find the equation of the line of intersection of the first two planes, then of the second and third planes, then find the intersection of these two lines, the intersection of the planes.
In practise, solve all three equations for one of the variables in terms of the other two.
Say $$\begin{aligned} c&=f_1(a,b) \\ c&= f_2(a,b) \\ c&=f_3(a,b) \end{aligned}$$
Set $f_1=f_2$ for the first line (aka equation in $a$ and $b$) and $f_2=f_3$ for the second.
Then find the intersection of these two lines in terms of $a$ and $b$ and substitute into (any) one of the $f_i$ to find the third coordinate of intersection.
Draw some pictures to understand.