Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finance A corporation borrowed \(1,500,000\) to expand its line of shoes. Some of the money was borrowed at \(3 \%,\) some at \(4 \%,\) and some at \(6 \% .\) Use a system of equations to determine how much was borrowed at each rate, given that the annual interest was \(74,000\) and the amount borrowed at \(6 \%\) was four times the amount borrowed at \(3 \% .\) Solve the system using matrices.

Step-by-Step Solution

Answer

The amounts borrowed at \(3\%, 4\%, 6\%\) are \(100000, 500000, 900000\), respectively.

Step 1: Formulate the System of Equations

Let \(x\), \(y\), and \(z\) be the amounts borrowed at the interest rates \(3\%\), \(4\%\), and \(6\%\), respectively. Three relationships can be formulated into equations as follows: \[x+y+z=1500000\] representing the total amount borrowed, \[0.03x+0.04y+0.06z=74000\] representing the total interest earned and \(z=4x\) representing the relationship between the amounts borrowed at \(6\%\) and \(3\%\).

Step 2: Express the system in matrix form

Using the coefficients of the variables from the system of equations, the matrix form can be represented as follows: \[\begin{pmatrix} 1 & 1 & 1 \ 0.03 & 0.04 & 0.06 \ -4 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix} =\begin{pmatrix} 1500000 \ 74000 \ 0 \end{pmatrix}\].

Step 3: Solve the System using Matrices

Use either matrix inversion or Gaussian elimination to solve the matrix system. Here, an example of solving the system using Gaussian elimination is shown: After applying Gaussian elimination, we can reduce the matrix to an upper triangular matrix, and hence determine the solutions: \(x = 100000, y = 500000, z = 900000\)

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with a common set of variables. Solving a system of equations means finding the values of the variables that satisfy all the equations simultaneously. In finance, systems of equations can be used to solve problems involving multiple unknowns, such as determining how much was borrowed at different interest rates in a loan situation. For instance, in the original exercise, we are given three equations representing the total amount borrowed, the total interest earned, and a specific relationship between amounts borrowed at particular rates. These equations are solved together to find the unknown amounts.

Matrix Representation

Matrix representation is a powerful mathematical tool for expressing and solving systems of equations. A matrix is a rectangular array of numbers arranged in rows and columns. In this context, matrices can succinctly represent systems of linear equations. Each row corresponds to an equation, and each column corresponds to a variable. The original exercise demonstrates how to convert a system of equations into a matrix form. This involves arranging coefficients of each variable in an orderly matrix. For example, coefficients of the variables from the system of equations are placed in the matrix \[ \begin{pmatrix} 1 & 1 & 1 \ 0.03 & 0.04 & 0.06 \ -4 & 0 & 1 \end{pmatrix} \], while the constants are in a separate matrix \[ \begin{pmatrix} 1500000 \ 74000 \ 0 \end{pmatrix} \].This matrix representation allows for systematic approaches like Gaussian elimination to solve for the variables.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations. It systematically transforms a matrix into a row-echelon form, making it straightforward to find the solutions. The method involves performing row operations to simplify the matrix such that it becomes upper triangular, where all elements below the main diagonal are zero. In the original exercise, Gaussian elimination is employed to reduce the matrix to find the values of variables. Once this upper triangular form is achieved, it is easy to perform back-substitution to solve for one variable at a time, leading to the solution

• \(x = 100,000\),
• \(y = 500,000\),
• \(z = 900,000\)

Thus, Gaussian elimination provides a step-by-step method for navigating through the complex matrix computations that arise from systems of equations.

Interest Rates in Finance

Interest rates are a fundamental concept in finance, indicating the cost of borrowing money. In problems involving loans, different interest rates can significantly affect the total interest paid over time. The original exercise involves a corporation borrowing at three different interest rates (3%, 4%, and 6%). Each rate affects the total interest accrued annually. Understanding the calculations of the interest amounts borrowed at each rate allows for effective financial planning.

• \(0.03x\) represents interest at 3%.
• \(0.04y\) represents interest at 4%.
• \(0.06z\) represents interest at 6%.

These kinds of exercises are typical in finance to determine optimal borrowing strategies and to ensure that all parts of a financial equation align perfectly with the given conditions.