Is there a way to find solution if all variables cancel out? 
*

*Factory A produces: 12 Tables and 6 Chairs per hour

*Factory B produces:  8 Tables and 4 Chairs per hour


How hours does EACH factory need to work to produce 48 Tables and 24 Chairs?
The linear equations are:
$$12A+8B=48$$
$$6A+4B=24$$
The first equation is twice the second, so everything cancels out.
But with trial and error, you find 3 solution:


*

*A=0 and B=6

*A=2 and B=3

*A=4 and B=0


What does it mean when variables cancel out or when all of them are zeroes?
 A: You actually won't ever stop finding solutions with trial and error.
The thing is that both equations carry the same information about what values the variables contain. Any solution such that:
$b = 6-(3a/2)$
or
$a = 4-(2b/3)$
is a valid solution.
For you question of how to find these solutions, you can find these equations by solving either of the two equations that you have for one of its variables. Then, you can generate as many solutions as you would like. For example, starting by solving for a variable:
$6a+4b=24$ -> $3a+2b=12$ -> $3a = 12-2b$ -> $a = 4 - 2b/3$
Then I just plug some number into b, for example, 7
$a=4-(2*7)/3$ -> $a = -2/3$
which means that one solution of the infinite amount of solutions, is: $a=-2/3$, and $b=7$
A: Your equations are correct. These equations are linear dependent. The first equation is twice of the second equation. That means that you one equation can be dropped. Now you can solve one of the equations for $A$ or $B$. Letßs solve the first equation for B.
$12A+8B=48\qquad |-12A$
$8B=48-12A \qquad|:8$
$B=6-1.5A$
If $A$ and $B$ need not to be integers then the solution is every point on the graph below, where $A,B \geq 0$.
If $A$ and $B$ must be integers then the three points (marked black) are the solutions. This is what you have already found out.

A: If both factories work together, in 1 hour they produce 20 tables and 10 chairs. Thus, it will require 2 hours and 24 minutes to produce required number of tables and chairs. ($20x=48, x=2.4$ hours, $10x=24, x=2.4$). I agree that the problem is ambiguous.
