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

Learn Before

Example

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 $b$ = the principal for the bank loan, and $f$ = the principal for the federal loan. Multiply Principal $\cdot$ Rate $\cdot$ Time ( $t=1$ ) to get the Interest. Organize into a table: / Loan / Principal ($) / Rate / Interest ($) / /---/---/---/---/ / Bank / $b$ / $0.105$ / $0.105b$ / / Federal / $f$ / $0.059$ / $0.059f$ / / 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 $b$ : $b = -f + 21{,}540$ Substitute into the second equation: $0.105(-f + 21{,}540) + 0.059f = 1{,}669.68$ $-0.105f + 2{,}261.7 + 0.059f = 1{,}669.68$ $-0.046f + 2{,}261.7 = 1{,}669.68$ $-0.046f = -592.02$ $f = 12{,}870$ To find $b$ , substitute $f = 12{,}870$ into the first equation: $b = -12{,}870 + 21{,}540$ $b = 8{,}670$ 6. Check the answer. $8{,}670 + 12{,}870 = 21{,}540$ $\checkmark$ $0.105(8{,}670) + 0.059(12{,}870) = 910.35 + 759.33 = 1{,}669.68$ $\checkmark$ 7. Answer the question. The principal for the bank loan was $\$8{,}670$ and the principal for the federal loan was $\$12{,}870$ .

Updated 2026-05-26

Contributors are:

Who are from:

References

OpenStax Intermediate Algebra

Learn Before

Related

Learn After