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 =
Step 3: Write out the equations based on the given information.
From the problem, we are given that the subsidized loan is $200 more than the combined total of the unsubsidized and parent loans. So, we can write the equation:
x = y + z + 200
We are also given that the unsubsidized loan is twice the amount of the parent loan. So, we can write the equation:
y = 2z
Now we have two equations:
x = y + z + 200
y = 2z
Step 4: Solve the system of equations to find the values of x, y, and z.
We can substitute the value of y from the second equation into the first equation to eliminate y:
x = (2z) + z + 200
Multiplying and combining like terms:
x = 3z + 200
Now we can substitute this expression for x back into the first equation to eliminate x:
(3z + 200) = (2z) + z + 200
Simplifying and combining like terms:
3z + 200 = 3z + 200
The z term cancels out, leaving us with:
200 = 200
This equation is always true, which means that there are infinitely many solutions. In this case, we can choose any value for z and then calculate the corresponding values for x and y.
For example, let's choose z = 100. Then, using the equation y = 2z:
y = 2(100) = 200
And using the equation x = y + z + 200:
x = 200 + 100 + 200 = 500
So, one possible set of loan amounts is:
Subsidized loan = $500
Unsubsidized loan = $200
Parent loan = $100
Remember, there are infinitely many solutions, so you can choose different values for z and calculate the corresponding values for x and y.