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Simultaneous equations help

Create simultaneous equations with the following information, and then solve them.

I am thinking of a three digit number. If you multiply the sum of its digits by 31 you get the original number. If you reverse the digits and add 99, you get the original number. Five times the middle digit is seven times the sum of the outside digits.

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1 Answer

Let the digits be a, b, and c, so our number is abc. 

Since it's a three digit number, we can represent the original number as (a*100) + (b*10) + c
(Why? Take the number "two hundred thirty seven" as an example. This equals 200 + 30 + 7, or 2 * 100 + 3 * 10 + 7. We apply the same concept to our original number)

So if we multiply the sum of the digits by 31, we get the original number. Therefore, 
31 * (a + b + c) = (a*100) + (b*10) + c.

If we reverse the digits (cba) and add 99, we get the original number, so
(c*100) + (b*10) + a + 99 = (a*100) + (b*10) + c

Five times the middle digit [b] is 7 times the sum of the outside digits [a and c], so
5b = 7(a + c)

Simplify each equation by distributing, etc as necessary and combining like terms, then solve by either elimination or by substitution. This should be enough to get you started.