SOLUTION: A+b=8,c-d=6,a+c=13,b+d=8. What are values of a,b,c and d

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Question 1043382: A+b=8,c-d=6,a+c=13,b+d=8. What are values of a,b,c and d
Found 2 solutions by solver91311, ikleyn:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Technically, there is no specific answer to your problem. That is because you have 5 variables and only 4 equations. is not now, never has been, and never will be the same thing as . However, presuming that you were simply being sloppy...
a + b = 8  =>  a = 8 - b
b + d = 8  =>  d = 8 - b, so the first thing we know is that a = d

a + c = 13 =>  a = 13 - c
c - d = 6  =>  d = c - 6
but since a = d
c - 6 = 13 - c  =>  2c = 19 =>  c = 9.5

You should be able to finish it yourself from here.

John

My calculator said it, I believe it, that settles it

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
A+b=8,c-d=6,a+c=13,b+d=8. What are values of a,b,c and d
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a + b        = 8,   (1)
       c - d = 6,   (2)
a +    c     = 13,  (3)
    b    + d = 8.   (4)

Add all four equations (1), (2), (3) and (4) (both sides). You will get

2a + 2b + 2c = 8 + 6 + 13 + 8 = 35,   or

a + b + c = 17.5.     (5)

Now distract eq.(1) from eq.(5). You will get

c = 17.5 - 8 = 9.5.

Next distract eq.(3) from eq.(5). You will get

b = 17.5 - 13 = 4.5.

Finally, from (1)  a = 8-b = 8-4.5 = 3.5,  and  from (4)  d = 8-b = 8-4.5 = 3.5.

Answer.  a = 3.5,  b = 4.5,  c = 9.5,  d = 3.5.

There is a (hidden) symmetry in these equations that helps to solve the system.

Close ideas work in other cases. See the lessons
    - The tricks to solve some word problems with three and more unknowns using mental math
    - Joint-work problems for 3 participants
in this site.