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Chapter 8: Problem 107

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.

Short Answer

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

Step by step solution

01

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%).
02

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 \(0.47x + 0.08y = 130500\) .
03

Matrix Formation

These simplified equations can be written in matrix form as follows: \[\begin{pmatrix} 5 & 1 \\ 0.47 & 0.08 \\ \end{pmatrix}\begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} 1500000 \\ 130500 \\ \end{pmatrix}\] .
04

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.

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Most popular questions from this chapter

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