Question 1043382
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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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<pre>
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.

<U>Answer</U>.  a = 3.5,  b = 4.5,  c = 9.5,  d = 3.5.
</pre>

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


Close ideas work in other cases. See the lessons 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/The-trick-to-solve-some-word-problems-with-three-and-more-unknowns.lesson>The tricks to solve some word problems with three and more unknowns using mental math</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Joint-work-word-problems-for-3-4-5-participants.lesson>Joint-work problems for 3 participants</A> 

in this site.