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- solve for x. we have x-8=x/3 +1/6.
- Now the first thing I want to do here
- There is multiple ways to do this problem,
- What I am gonna do is to simplify the fractions.
- I am going to multiply everything times the least common multiple of all
- of these guys denominator.
- This is essentially,
- x over 1, this is 8 over1,
- x over 3,
- 1 over 6
- The least common multiple of 1, 3 and 6 is 6.
- So if I multiply everything times 6,
- it's going to clear out these fractions.
- so, these weren't fractions to begin with
- So just multiplying them by 6,
- so it becomes 6x minus 6 times negative 8 or 6 times 8 is 48
- and there is , we are subtracting it right over there,
- And then we have x over 3 times 6, let me just write out here
- So that's going to be 6 times x over 3, plus,
- 6 times 1 over 6 .
- Or we get, 6x minus 48 is equal to
- 6 times something divided by 3,
- that's the same thing as 6 divided by 3 times that something
- that's just going to be equal to 2x plus
- 6 times one-sixth or 6 divide by 6 is just going to be 1
- So that first step cleared out all of the fractions
- and now this is just a straight- forward problem
- with all integer co-efficients or integers on
- either side of the equation.
- What we wanna do is, we wanna isolate all of the x
- on one side or the other.
- we might as well isolate them all on the left hand side
- So,lets subtract 2x from both sides,
- we want to get rid of this 2x here.
- That's why I am subtracting the 2x.
- So let's subtract 2x from both sides, and,
- on the right hand side, I have 2x plus 1 minus 2x
- those cancelled out.
- That was the whole point
- So , I'm left with just
- this 1 over here
- On the left hand side, I have 6x minus 2x ,
- Well, that's just going to be 4x if I
- have 6x of something minus 2 of that something,
- I have 4 of that something.
- minus 48. And now I can.. lets see I want to get rid of
- this 48 on the left hand side, I want only x here.
- So let me add 48 to both sides of the equation.
- this in a new color! So let me add 48
- to both sides of this equation.
- And on the left hand side, 4x minus 48 plus 48
- those cancelled out, I am left with just a 4x
- And on the right hand side, 1 plus 48 is going to be
- 49. And now I have isolated the x,
- but still multiplied by a 4 so to make that
- a 1 co-efficent, lets multiply both sides by one-fourth
- or you can also that
- lets divide both sides by 4.
- Anything you do the one side , you have to do to the other
- And so you have.. What do we have over here,
- 4x over 4 is just x,
- x is equal to 49 over 4.
- That's about as far as we can simplify it
- because these don't have any common factors, 49 and 4.
- Lets check to see whether 49 by 4 is indeed the answer.
- So, let us put it into the original equation
- well the original equation is what we have in green here.
- before we multiplied it,
- before we multiplied by 6.
- But in theory we should be able to put it into any of these steps
- and x should satisfy it.
- Lets do it in our original equation.
- So we have x minus 8, so we have
- 49 over 4 minus 8 should be
- equal to 49 over 4 over 3 plus 1 over 6.
- So lets see what we can do here, so what we can..
- lets see.. we can multiply, well like we did before.
- We can multiply both sides of this equation by 6,
- that will help simplify a lot of the fractions
- here. So if we multiply both sides of this equation
- by 6, so we are going to multiply everything
- .. everything by 6.
- what do we get,
- On the left hand side , 6 over 4 is the same thing as 3 over 2
- So this is going to be 3 times 49 over 2,
- 3 times 49 over 2 minus 48
- will be equal to
- 6 divided by 3 is going to be 2, so its
- 2 times 49 over 4 , which is the same thing as
- 49 over 2 , and then 6 times one-sixth is plus 1
- and lets see , we can, well I am just going to, I am just going to
- actually just evaluate them out. I could just essentially have to
- you know subtract 49 over 2 from both sides, that will simplify
- things, but let me just, let me just figure out what these
- evaluate to. So 3 times 49, 49 times 3
- you could think about it's going to be 3 less than 50 times 3
- so its 147. But less just multiply out 9 times 3 is 27,
- 4 times 3 is 12, plus 2, is 14. So 147.
- This is 147 over 2.
- and then lets put this over a denominator of 2.
- So 48 is equal to 96 over 2, right
- 96 divide by 2 is 48. I just multiplied this by 2.
- So this is minus 96 over 2, needs to be equal to
- 49 over 2 plus, and instead of having this 1,
- lets write that as 2 over 2.
- Now what is 47 minus 96.
- So 47 minus a hundred would be,
- 147 minus a 100 would be 47,
- we are going to subtract 4 more than that.
- So we are going to have, it's going to be..oh sorry!
- We are going to subtract 4 less than a 100
- So it's going to be 147 minus 96 is going to be 51,
- 51 over 2 is equal to 49 plus 2 is 51 over 2.
- so it all checks out.

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