Problem 107 - Precalculus with Limits

Question

FINANCE A small shoe corporation borrowed \(\$ 1,500,000\) to expand its line of shoes. Some of the money was borrowed at \(7 \%,\) some at \(8 \%,\) and some at 10\(\% .\) Use a system of equations to determine how much was borrowed at each rate if the annual interest was \(\$ 130,500\) and the amount borrowed at 10\(\%\) was 4 times the amount borrowed at 7\(\% .\) Solve the system using matrices.

Step-by-Step Solution

Answer

Write your obtained answers for x, y, and z here.

Step 1: Setup of Equations

There are three types of loans: at 7%, 8%, and 10%. Denote the amounts borrowed at each rate as \(x\), \(y\), and \(z\) respectively. From the problem, we have three equations. The first being \(x + y + z = 1500000\) (the total sum borrowed), the second equation being \(0.07x + 0.08y + 0.1z = 130500\) (the total interest combined) and the third equation \(z = 4x\) (amount borrowed at 10% is four times the amount borrowed at 7%).

Step 2: Substitution & Simplification

Substitute the third equation into the other two to simplify them. So, \(x + y + 4x = 1500000\), this simplifies to \(5x + y = 1500000\). Second equation becomes \(0.07x + 0.08y + 0.4x = 130500\), which simplifies to \(0.47x + 0.08y = 130500\) .

Step 3: Matrix Formation

These simplified equations can be written in matrix form as follows: \[\begin{pmatrix} 5 & 1 \\ 0.47 & 0.08 \\ \end{pmatrix}\begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} 1500000 \\ 130500 \\ \end{pmatrix}\] .

Step 4: Solving the Matrix

Use an appropriate method like Gaussian Elimination or Cramer's Rule to solve this matrix. Solving this can give us the values for x and y. Once these are obtained, substitute x into \(z = 4x\) to find the value for z.