Rachel has 60 scones. she sells them to robbie, cameron, louis, tom and charlie in that order. each customer buys more scones then the last and the increase is the same number in each case. robbie and cameron's combined total number of scones is three sevenths of the total of louis, tom and charlie. how many scones does each boy buy?

We will use the variable r to represent the number of scones sold to Robbie.  We will use the variable x to represent the increase in scones sold to each person (since the increase is the same, we can use the same variable each time).
Scones sold to Cameron = r + x.
Scones sold to Louis = the scones sold to Cameron + x = r + x + x = r + 2x.
Scones sold to Tom = scones sold to Louis + x = r + 2x + x = r + 3x.
Scones sold to Charlie = scones sold to Tom + x = r + 3x + x = r + 4x.
We add all of these together to get our total of 60:
r + (r + x) + (r + 2x) + (r + 3x) + (r + 4x) = 60
Combine our like terms and we have 5r + 10x = 60 (there are 4 r's and 10 total x's).
We have another equation involved here.  We know that Robbie + Cameron, or r + r + x = 2r + x, is 3/7 of the total of the other three boys (this total given by r + 2x + r + 3x + r + 4x = 3r + 9x).  We can write this as:
2r + x = 3/7(3r + 9x)
We can cancel the 7.  Fraction bars are the same as division, so we cancel by multiplying:  7(2r + x) = 3(3r + 9x), which simplifies to 14r + 7x = 9r + 27x after using the distributive property.
In order to work with a system of equations (two equations at the same time) it's easier if they look the same.  Let's get r and x on one side of the equation.  We will start by cancelling 9r.  Since it's positive, we can cancel or move it by subtracting:  14r + 7x - 9r = 9r - 9r + 27x, which gives us 5r + 7x = 27x.  Now we want to move the 27x.  Since it's also positive we will subtract it:
5r + 7x - 27x = 27x - 27x, which gives us 5r -20x = 0.
Let's align them vertically:

The coefficients of our r's are the same, so we will eliminate those by subtracting:

Now that our r's are gone, we have 30x (10x - -20x = 30x) and 60 on the other side:
30x = 60
Divide both sides by 30:

Plug this into our first equation, 5r + 10x = 60:

Subtract 20 from both sides:

Divide both sides by 5:

This tells us that r, the scones sold to Robbie, were 8 and x, the change between the number of scones sold to each person, was 2.  That means she sold 8 to Robbie, 10 to Cameron, 12 to Louis, 14 to Tom and 16 to Charlie.