Question 615967
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We're trying to help our {{{highlight(cross(soon))}}} <U>son</U> with his math homework but we're totally stuck one this problem:
A store has a sale on almonds, pecans and pistachios. One lb of almonds, one lb of pecans and one lb of pistachios 
cost $12. Two lbs of almonds and three lbs of pecans cost $16. Three lbs of pecans and two lbs pistachios cost $24. 
Find the price of each kind of nut.

Thank you in advance for any help.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The solution by @Theo is tedious and boring, and you're unlikely to learn anything useful from his post.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Meanwhile, the problem has a much simpler solution, which teaches you to think and to approach the problem 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in a new way. This solution is effective and instructive, and can work in many other problems.



<pre>
Let  'a'  be the pounds of almonds,
     'b'  be the pounds of pecans, and
     'c'  be the pounds of pistachios.


You have 3 equations that need to be solved simultaneously.

     a +  b + c  = 12,     (1)
    2a + 3b      = 16,     (2)
         3b + 2c = 24.     (3)


This system of equations has a hidden symmetry, which will help us to solve.


Add equations (2) and (3)  (both sides separately).  You will get

    2a + 6b + 2c = 40.     (4)


From equation (4), subtract equation (1), multiplied by 2.  
The terms with  'a'  and  'c'  will cancel, and you will get

    6b - 2b = 40 - 2*12,

or

       4b = 16.


Hence,  b = 16/4 = 4.



Now, substitute b=4 into equation (2) and find  'a'

    2a + 3*4 = 16  --->  2a = 16 - 12 = 4  --->  a = 4/2 = 2.



Next, substitute b=4 into equation (3) and find  'c'

    3*4 + 2c = 24  --->  2c = 24 - 12 = 12  --->  c = 12/2 = 6.


At this point, the problem is solved completely.


<U>ANSWER</U>.  a = $2 per pound (almonds);  b = $4 per pound (pecans),  and  c = $6 per pound (pistachios).
</pre>

Solved.

<H3>The lesson to learn</H3

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If you have to solve a system of three equations with three unknowns, 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;look if there is a hidden symmetry in it that could help.