1Cademy - Solving a Dual Student Loan Interest Application Using a System of Equations

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Solving a Dual 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: 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? 1. Read the problem. 2. Identify what to find: the principal of each loan. 3. Name the unknowns. Let $b$ = the principal for the bank loan, and $f$ = the principal for the federal loan. Organize into a table: / Loan / Principal ($) / Rate / Interest ($) / /---/---/---/---/ / Bank / $b$ / $0.025$ / $0.025b$ / / Federal / $f$ / $0.069$ / $0.069f$ / / Total / $18{,}000$ / / $1{,}066$ / 4. Translate into a system of equations: $\left\{\begin{array}{l} b + f = 18{,}000 \ 0.025b + 0.069f = 1{,}066 \end{array} ight.$ 5. Solve the system using substitution. Solve the first equation for $b$ : $b = -f + 18{,}000$ Substitute into the second equation: $0.025(-f + 18{,}000) + 0.069f = 1{,}066$ $-0.025f + 450 + 0.069f = 1{,}066$ $0.044f + 450 = 1{,}066$ $0.044f = 616$ $f = 14{,}000$ To find $b$ , substitute $f = 14{,}000$ into the first equation: $b + 14{,}000 = 18{,}000$ $b = 4{,}000$ 6. Check the answer. $4{,}000 + 14{,}000 = 18{,}000$ $\checkmark$ $0.025(4{,}000) + 0.069(14{,}000) = 100 + 966 = 1{,}066$ $\checkmark$ 7. Answer the question. The principal for the bank loan was $\$4{,}000$ and the principal for the federal loan was $\$14{,}000$ .

Updated 2026-05-26

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OpenStax Intermediate Algebra

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