They're two separate equations - the 500 isn't "changing" into 60. The first equation deals with the value of the coins. $5.00 is the total value, so that's been changed to 500 cents. n_1 is the number of pennies, so it simply has a value of n_1 cents. n_10 is the number of dimes, so it has a value of 10 * n_10, and likewise for n_25 and quarters.
The second equation deals with the total number of coins, which you know to be 60, and you know it is the sum of the number of pennies, dimes, and quarters, hence n_1 + n_10 + n_25 = 60.
Is it possible to have 60 coins, made up of pennies, dimes and quarters
that add up to $5.00? Justify your answer.
Let = number of pennies,
. . . = number of dimes,
. . . = number of quarters.
where are nonnegative integers.
There are 60 coins: .
Their value is 500 cents: .
Subtract  from : .
So we have: .
The left side is a multiple of 3 . . . The right side is not.
. . There are no solutions in integers.
Therefore, the task is impossible.