Question

Ok so I have this problem, 12x+24=6x+3. WHAT DO I DO IF THE X HAS DIFFERENT NUMBERS. Oh and for 5-2x=2x-5 what do i do if ones a negative and ones a positive. Im trying to find the answer and if its 1 solution, no solution, or infinite solutions. Im sure people on here can reply quicker then my math teacher.

Answer 1

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.