Loan Mixture: A student takes out a total of $5000 in three loans: one subsidized, one unsubsidized, and one from the parents of the student. The subsidized loan is $200 more than the combined total of the unsubsidized and parent loans. The unsubsidized loan is twice the amount of the parent loan. Find the amount of each loan.

Step 1:

x = subsidized
y = unsubsidized
z = parent

Step 2:

x + y+ z = 5000

Step 3:

Solution:

Subsidized loan =
Unsubsidized loan =
Parent loan =

Get everything in terms of one variable

If the parent loan is Z
the unsubsidized loan is 2Z
the subsidized loan is 200 + Z + 2Z
5000=Z + 2Z + 200 + Z +2Z solve for Z
5000=6Z + 200
4800=6Z
800=Z
The parent loan is 800
The unsubsidized loan is 1600
The subsidized loan is 2600

Step 3:

Based on the given information, we can set up a system of equations to solve for the unknowns.

Since the subsidized loan is $200 more than the combined total of the unsubsidized and parent loans, we can write the equation as:
x = y + z + 200 (equation 1)

Since the unsubsidized loan is twice the amount of the parent loan, we can write the equation as:
y = 2z (equation 2)

We also know that the total amount of the loans is $5000, so we can write the equation as:
x + y + z = 5000 (equation 3)

Now we can solve this system of equations to find the values of x, y, and z.

Solution:

Substituting equation 2 into equation 1, we get:
x = 2z + z + 200

Simplifying this equation, we have:
x = 3z + 200 (equation 4)

Now we can substitute equations 2 and 4 into equation 3 to solve for z:
(3z + 200) + (2z) + z = 5000

Combining like terms, we get:
6z + 200 = 5000

Subtracting 200 from both sides of the equation, we have:
6z = 4800

Dividing both sides of the equation by 6, we get:
z = 800

Now we can substitute the value of z into equation 2 to find y:
y = 2z = 2 * 800 = 1600

Lastly, we can substitute the values of z and y into equation 1 to find x:
x = y + z + 200 = 1600 + 800 + 200 = 2600

Therefore, the amount of each loan is as follows:
Subsidized loan = $2600
Unsubsidized loan = $1600
Parent loan = $800