Confused with Rate Problems/Coefficient, Pos/Neg signs.

[Voltux]

How do I determine if I subtract or add the coefficient to the fraction?
ex. 1-(-48/18)*Sh=384 vs 1+(-48/18)*Sh=384? Also, It would be the Sh+Sm=312 where I get the coefficient "from" right, specifically the second "Sh" term where as the sm term is eliminated with -sm.

Thanks!

(The problem above may not match exactly the problem below, I was trying to copy it back from my notes, Taken from KhanAcademy)

Problem:
Sh(Speed Horse)=18mph
Sm(Speed Motorcycle)=48mph
Dt(Distance Total)=312Miles
Tt(Total Time)=8Hours
Solve for Miles Traveled By Horse

(Dh/18)+(Dm/48)=8Hrs
Dh+Dm=312Mi

1. Remove speed by Motorcycle

-48*(Dm/48)+(Dh/18)=8*-48 -> -Dm(-/+?)(-48/18)*Dh=-384

2. Add to Dm+Dh=312mi

-Dm(-/+?)(-48/18)*Dh=-384 + (Dm+Dh=312)

-Dm +Dm cancel out

(-/+?)(-48/18)*Dh=-384

Subtract 312 from 384, retain negative status.

(-/+?)(-48/18)*Dh=-72

Dh from equation becomes coefficient 1 or 18/18 for the fraction.

18/18(-/+?)(-48/18)*Dh=-72

3. Add/Subtract the Fractions and Multiply inverse by -72

18-(-48) = 66
18+(-48)= -30

Dh=-72/1*18/66 = - 19 7/11
Dh=-72/1*18/-30 = 43 1/5

I can easily check to see which one fits the answer and then pick the correct one however it's lots of extra work...So I'm wondering how to pick the correct sign for the answer?

For sake of completeness to check the answer:

4. Dh-Dt= 268 4/5, Then take 268 4/5 divide that by Speed Motorcycle (48) That gives you 5 3/5 Hours travel.

Then you take 5 3/5 Hours + Dh/18 gives you 2 2/5

If 5 3/5 + 2 2/5 = Tt (8 Hours) Then you have your correct answer.

Thanks for your time, and I really look forward to putting this in the past, I've spent waaay too much time on these problems in the last week :)

 

[symbolipoint]

How do I determine if I subtract or add the coefficient to the fraction?
ex. 1-(-48/18)*Sh=384 vs 1+(-48/18)*Sh=384? Also, It would be the Sh+Sm=312 where I get the coefficient "from" right, specifically the second "Sh" term where as the sm term is eliminated with -sm.

The expressions contain a factor on the Sh term, (-48/18). The additive inverse of (48/18) is (-48/18). Also, the additive inverse of (-48/18) is (48/18).

Note that (48/18) + (-48/18) = 0.
Also be aware that 1 + (48/18) will simply be the sum of these two numbers, while
1 + (-48/18) will point the second number of the sum in the OPPOSITE direction, meaning that
1 + (-48/18) = 1 - (48/18).

Based on that, what is the proper understanding of 1-(-48/18)*Sh ?