Solving a Student Loan Interest Application Using a System of Equations
Apply the seven-step problem-solving strategy and a mixture table to find the principal given the amount of interest earned. Problem: Rosie owes $21{,}540 on her two student loans. The interest rate on her bank loan is 10.5% and the interest rate on the federal loan is 5.9%. The total amount of interest she paid last year was $1{,}669.68. What was the principal for each loan? 1. Read the problem. A chart will help organize the information. 2. Identify what to find: the principal of each loan. 3. Name the unknowns. Let = the principal for the bank loan, and = the principal for the federal loan. Multiply Principal Rate Time () to get the Interest. Organize into a table: / Loan / Principal ($) / Rate / Interest ($) / /---/---/---/---/ / Bank / / 0.105 / 0.105b / / Federal / / 0.059 / 0.059f / / Total / 21{,}540 / / 1{,}669.68 / 4. Translate into a system of equations from the Principal and Interest columns: \left\{\begin{array}{l} b + f = 21{,}540 0.105b + 0.059f = 1{,}669.68 \end{array} ight. 5. Solve the system using substitution. Solve the first equation for : Substitute into the second equation: 0.105(-f + 21{,}540) + 0.059f = 1{,}669.68 -0.105f + 2{,}261.7 + 0.059f = 1{,}669.68 To find , substitute into the first equation: 6. Check the answer. checkmark 0.105(8{,}670) + 0.059(12{,}870) = 910.35 + 759.33 = 1{,}669.68 checkmark 7. Answer the question. The principal for the bank loan was $8{,}670 and the principal for the federal loan was $12{,}870.
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Learn After
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