9.) Using elimination:
3x+6y=27
x+2y=11
The idea of elimination is to eliminate (cancel out) a variable so that only one variable exists. For example, if x is eliminated, then y is being solved, and if y is eliminated, then x is being solved. We must pick a variable in the system (x or y) and find a LCM that is the additive inverse, meaning the variables cancel out. Some example of an additive inverse: -5+5=0 and 6-6=0; it is the inverse (opposite) of addition (so subtraction) in which undoes the operation to cancel it out to zero. Likewise for subtraction to addition. So, let’s choose a variable, identify the LCM of the two same variables in both equations, and either negate or add the variables together so they cancel out to 0.
Let’s choose x:
The LCM of 3 and 1 is 3, so we only need to multiply the second equation by a factor. 1•-3=-3, so we’ll multiply the entire second equation by 3, and 3-3=0:
3x+6y=27
-3(x+2y)=-3(11)
Distribute and multiply:
-3x-6y=-33
Here is the new set:
3x+6y=27
-3x-6y=11
Now, simply add or subtract the two equations:
(3x-3x)+(6-6)=27+11
Simplify:
0+0=38
0=38
This is a false statement, and thus the system does not have any solutions, meaning the lines are parallel.
So, the answer is: No Solutions