Equation with fractions and variables on both sides

[KeenDave]

Hello.

I’ve gotten to the part where I want to eliminate the fractions on each side. And I do realize the best number to multiply each side with is 8.

(3/4)x + 2 = (3/8)x - 4

= 8((3/4)x+2) = 8((3/8)x-4)

The solution I found first divides the 8 by 4 to get 2. No prob with that. However, it goes on then take that 2 and multiply it by 3 as opposed to multiplying 8 by 3 to get 24 and then 24/2 which equals 12. I understand I’m wrong here but why is my thought process wrong?

 

[khansaheb]

= 8((3/4)x+2) = 8((3/8)x-4)

First the first "=" sign should not be there. So you should have:

8 * (3/4 * x + 2) = 8 * ( 3/8 * x - 4).........that becomes (distributing 8 over the first parenthesis and the second one separately)

2 * 3 * x + 8 * 2 = 3 * x - 4 * 8 .........that becomes

6 * x + 16 = 3 * x - 32 .........that becomes

6* x - 3 * x = -32 -16 ...... and continue......

then 24/2 which equals 12

Where did this /2 come from? I think you are skipping steps and confusing yourself.

 

[Dr.Peterson]

However, it goes on then take that 2 and multiply it by 3 as opposed to multiplying 8 by 3 to get 24 and then 24/2 which equals 12.

All you are doing here is multiplying 8 * 3/4. You can do that either by multiplying 8 * 3 and then dividing by 4 (not 2) to get 24/4 = 6, or by first dividing 8 by 4 and then multiplying by 3, to get 2 * 3 = 6. Same result either way.

The two methods correspond to multiplying 8/1 * 3/4 directly and then simplifying, or simplifying first and then multiplying.

Or, equivalently, to thinking of it as (8 * 3) * 1/4 vs (8 * 1/4) * 3.

 

[KeenDave]

First the first "=" sign should not be there. So you should have:

8 * (3/4 * x + 2) = 8 * ( 3/8 * x - 4).........that becomes (distributing 8 over the first parenthesis and the second one separately)

2 * 3 * x + 8 * 2 = 3 * x - 4 * 8 .........that becomes

6 * x + 16 = 3 * x - 32 .........that becomes

6* x - 3 * x = -32 -16 ...... and continue......


Where did this /2 come from? I think you are skipping steps and confusing yourself.

Thank you!! I sure did confuse myself! Haha I somehow thought I could multiply the 8 across after already having simplified the fraction (8/4=2). After this step the 8 no longer existed. So glad I asked! I appreciate your response!

 

[KeenDave]

All you are doing here is multiplying 8 * 3/4. You can do that either by multiplying 8 * 3 and then dividing by 4 (not 2) to get 24/4 = 6, or by first dividing 8 by 4 and then multiplying by 3, to get 2 * 3 = 6. Same result either way.

The two methods correspond to multiplying 8/1 * 3/4 directly and then simplifying, or simplifying first and then multiplying.

Or, equivalently, to thinking of it as (8 * 3) * 1/4 vs (8 * 1/4) * 3.

Thank you so much! I managed to confuse myself here lol. Tried multiplying the 8 across after having already divided it by the 4-denominator lol. It’s so painfully obvious now that you pointed out the 8*3/4. Thanks again!