Q +  H =   30     (1)   (counting coins)    Q stands for quarters
25Q + 50H = 875     (2)   (counting cents)    H stands for half-dollars


In eq(2) cancel the factor 25 in both sides. You will get the modified system in the form

  Q +  H =   30     (1')
  Q +  2H =  35     (2')

Next subtract eq(1') from eq(2') (both sides).  You will get

       H = 35 - 30 = 5.


Answer.  5  half-dollars and 30-5 = 25 quarters.

Let assume for a minute that all 30 coins are 25-cent quarters.

Then the total would be 30*25 = 750 cents.

But in reality we have 875 cents.

It because we counted half-dollars as quarters.

Now, let start return back these half-dollars.

With each returned half-dollar we diminish the difference of 875-750 = 125 cents by 25 cents.

How many half-dollars should we return ? - But of course,  = 5 half-dollars.


We completed our calculations and got the same answer.