Ten brothers receive 100 gold coins between them . each brother receives a constant number more than the next oldest . the seventh oldest brother receives 7 gold coins . how much does each brother receive ? show all your calculations

Let's set 'a' as the coins the first brother recieves (the youngest) and for every older brother, we will add 'x' (the constant number)
Since we know the sum has to be equal to 100, our equation is:
100=a+(a+x)+(a+2x)+(a+3x)+...+(a+8x)+(a+9x)

the 7th oldest brother is the 3rd youngest, therefore, if we look at our equation, he gets (a+2x) - we know now that 7=a+2x

We can now simplify the first equation by summing up our 'x' and 'a'
we get: 100=10a+45x

In order to resolve the equation (since we have 2 variables), we can use the subtraction method

  100=10a+45x 
-10(7=a+2x) 

  100=10a+45x 
-(70=10a+20x) 

30=25x
1.2 = x

and for 'a' ->  7 = a +2(1.2)
a=4.6

Therefore, the youngest gets 4.6 coins (a), the second 5.8 (a+x) ...