Answer: one solution; that solution is x = -7/2 which is the same as x = -3.5
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Work Shown:
The idea is to get all the x terms to one side, and the other stuff to the other side. We follow PEMDAS in reverse to help isolate x
12x+24 = 6x+3
12x+24-24 = 6x+3-24 .... subtract 24 from both sides
12x = 6x-21
12x-6x = 6x-21-6x ...... subtract 6x from both sides
6x = -21 ...... now the x terms are all on one side
6x/6 = -21/6 ...... dividing both sides by 6
x = -21/6
x = (-7*3)/(2*3)
x = -7/2
x = -3.5
This is the only solution. Whenever we have different coefficients for the x terms like this, we can see right off the bat we have exactly one solution. Consider graphing y = 12x+24 and y = 6x+3. The two lines will intersect at the point (x,y) where the x coordinate is the solution to the original equation. There is only one point of intersection which corresponds to exactly one solution.
The thing to notice about y = 12x+24 and y = 6x+3 is that the slopes are not the same, so the lines aren't parallel. Parallel lines always have equal slopes but different y intercepts. If your teacher gave you 12x+24 = 12x+3, then the two lines to graph would be y = 12x+24 and y = 12x+3. At this point, we would get two parallel lines that don't intersect, and therefore we wouldn't get any solutions here.
