Sophie has $4.20 worth of dimes and quarters. She has twice as many quarters as dimes. Write a system of equations that could be used to determine the number of dimes and the number of quarters that Sophie has. Define the variables that you use to write the system.

Let's start solving this problem step by step.

Step 1: Define the variables.
Let's say the number of dimes is 'd' and the number of quarters is 'q'.

Step 2: Write the equation for the total value of dimes.
Since each dime is worth $0.10, the total value of dimes can be calculated by multiplying the number of dimes by $0.10:
Total value of dimes = 0.10 * d

Step 3: Write the equation for the total value of quarters.
Since each quarter is worth $0.25, the total value of quarters can be calculated by multiplying the number of quarters by $0.25:
Total value of quarters = 0.25 * q

Step 4: Write the equation for the given information.
We are given that Sophie has twice as many quarters as dimes. Mathematically, this can be expressed as:
q = 2d

Step 5: Write the equation for the total value of dimes and quarters.
The total value of dimes and quarters is equal to $4.20. Mathematically, this can be expressed as:
Total value of dimes + Total value of quarters = $4.20

Now we can combine all the equations we derived:
0.10 * d + 0.25 * q = 4.20 (Equation 1)
q = 2d (Equation 2)

These equations form a system of equations that can be used to determine the number of dimes and the number of quarters that Sophie has.

q = 2d

10d+25q = 420