George has 21 coins with a total value of $12.75. He knows he only has quarters and dollar coins in his pocket. How many coins of each type does he have? Create a system to solve.

number of $1 coins --- x

number of quarters ---- y

x + y = 21

100x + 25y = 1275 --> 4x + y = 51

subtract them:
3x = 30
x = 10
then y = 11

He has 10 dollar coins and 11 quarters

check
value: 100(10) + 25(11) = 1275 cents = $12.75

To solve this problem, we can create a system of equations based on the given information.

Let's assume the number of quarters George has is represented by 'q' and the number of dollar coins is represented by 'd'. We can express the total value of the coins and the total number of coins using the equations:

Equation 1: 0.25q + 1.00d = 12.75 (represents the total value of the coins)
Equation 2: q + d = 21 (represents the total number of coins)

Now, we have a system of two equations with two variables. We can solve this system either by substitution or elimination method.

Let's solve it using the substitution method:

From Equation 2, we can express q in terms of d:
q = 21 - d

Now, substitute this expression for q in Equation 1:
0.25(21 - d) + 1.00d = 12.75

Simplify the equation:
5.25 - 0.25d + 1.00d = 12.75
0.75d = 7.50

Divide both sides by 0.75:
d = 10

Now substitute this value of d back into Equation 2 to find the value of q:
q + 10 = 21
q = 11

Therefore, George has 11 quarters and 10 dollar coins.

We have solved the problem by setting up a system of equations and using the substitution method to find the values of q and d that satisfy both equations.