Question

Translate to a system of equations and solve:

Laura owes \(18,000 on her student loans. The interest rate on the bank loan is 2.5% and the interest rate on the federal loan is 6.9 %. The total amount of interest she paid last year was \)1,066. What was the principal for each loan?

Short answer

The principal amount on bank loan is $4000 and on federal loan is $14000.

Step-by-step solution

Step 1. Given Information    

The given data is that Laura owes $18,000 on her student loans. The interest rate on the bank loan is 2.5% and the interest rate on the federal loan is 6.9 %. The total amount of interest she paid last year was $1,066.

Step 2. Explanation     

Let x be the principal on the bank loan and ybe the principal on the federal loan.

Total loan owes is $18000 that is $x+y=18000--\left(1\right)$

The total amount of interest she paid is $1066.

As we are given the interest rates of both the loans thus, the expression is equal tolocalid="1647685834075" $0.025x+0.069y=1066--\left(2\right)$

Step 3. Calculation 

Multiply equation (1) with $0.069$and write the revised equation.

$$\begin{gathered} 0.069(x+y)=0.069\left(18000\right) \\ 0.069x+0.069y=1242--\left(3\right) \end{gathered}$$

Solve the equations (2) and (3) by subtracting equation (2) from equation (3) to find the value of x.

$$\begin{gathered} 0.069x+0.069y-0.025x-0.069y=1242-1066 \\ 0.044x=176 \\ x=4000 \end{gathered}$$

Substitute the value of x in equation (1) to find the value of y.

$$\begin{gathered} 4000+y=18000 \\ y=14000 \end{gathered}$$