How can we do this quickly?

Question

Hello everyone.

For years, everyone's been talking about plugging in numbers for algebra problems. You pick a number for the variable, plug it in, get an answer, and then find that answer in the answer choices.

I think this is dangerous, because it only works sometimes, and I think that algebra or some other trick is always important as a backup. Plugging in should be done by people who understand it and when not to use it.

Sorry for waxing on about that, but I ran across a problem today that would be very difficult to plug numbers into, but is also very time consuming using straightforward algebra. You might expect it to work out well with substitution, but you'll see that it doesn't. So I thought I'd post it here and see what everyone thought about it, and if there's some clever thing that I'm not seeing. 

--------------
If 2x + 3y = 1, what is (x/2) + (y/3) in terms of y?

A) y/5

B) (1 - 3y)/2

C) (1 - 3y)/4

D) (3y + 4)/15

E) (3 - 5y)/12

smcgrath12 · Answer

I guess I am losing my mind. But for (x/2) + (y/2), I am getting (1-y)/4. This is not an option listed.

ywilfred · Answer

i'm getting (1-y)/4 as well

ian7777 · Answer

Sorry, my bad completely. The question should read, 

"...what is (x/2) + (y/3) in terms of y?"

I'll fix the original post.

carsen · Answer

If 2x + 3y = 1, what is (x/2) + (y/3) in terms of y?

---> 2x+3y=1
---> 2x=1-3y
---> x= (1-3y)/2
---> x/2=(1-3y)/4

Add y/3 on both sides.....

(x/2)+(y/3)= (1-3y)/4 + (y/3)

---> u get (3-5y)/12

(Solved less than one minute)

ywilfred · Answer

(E). 20 secs. I think good old substitution is still very fast in this case. If you note that you'll end up with 2 and 3 as denominator, you can work two steps ahead while substituting by adjusting the integer value of y as you work throuhg the problem.

hardworker_indian · Answer

I guess solving might be faster than substituting numbers (I mean, if Ian hadn't given the preface).

Let the value be M

2x + 3y = 1
3x + 2y = 6M

Solving this,
-5y = 12M-1
M = (3-5y)/12

rahul · Answer

15 seconds , I got it as E . 

divide the existing equation by 4 and move 3y/4 to the right side , 
you know x/2 and then simply add y/3 .

x /2 = (1-3y)/4

x/2 + y/3 = (1-3y)/4 + y/12
 = (3-5y)/12

rahul · Answer

Infact the moment you see the denominator as 12 , you can choose the choice E , because there is no other choice which has denominator 12 .

My bad , if i had not overlooked the choices , i think this problem can be solved with in 15 seconds .

longhorn2020 · Answer

This is probably one of the most straightforward way of solving the equation!

How can we do this quickly?

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