Problem 108 - Precalculus with Limits

Question

FINANCE A small software corporation borrowed \(\$ 500,000\) to expand its software line. Some of the money was borrowed at \(9 \%,\) some at \(10 \%,\) and some at 12\(\% .\) Use a system of equations to determine how much was borrowed at each rate if the annual interest was \(\$ 52,000\) and the amount borrowed at 10\(\%\) was 2\(\frac{1}{2}\) times the amount borrowed at 9\(\%\) . Solve the system using matrices.

Step-by-Step Solution

Answer

The small software corporation borrowed $90,000 at 9%, $250,000 at 10%, and $182,000 at 12%

Step 1: Define your variables

Let's define the variables: \( x \) is the amount borrowed at 9%, \( y \) the amount at 10%, and \( z \) the amount at 12%.

Step 2: Formulate the equations

Based on the information given in the problem, a system of three equations can be formulated: 1. The total amount borrowed: \( x + y + z = 500,000 \)2. The total interest: \( 0.09x + 0.10y + 0.12z = 52,000 \)3. The relation between the amount borrowed at 9% and 10%: \( y = 2.5x \)

Step 3: Alter original equations

To transform this system into martrix for easier solving substitute the third equation into the first two. The equations now become: 1. \( x + 2.5x + z = 500,000 \)2. \( 0.09x + 0.10 * 2.5x + 0.12z = 52,000 \)

Step 4: Set up matrix

Convert these 2 equations into a matrix:\[\begin{{bmatrix}}3.5 & 1\0.345 & 0.12\\end{{bmatrix}} * \begin{{bmatrix}}x\z\end{{bmatrix}} = \begin{{bmatrix}}500,000\52,000 \end{{bmatrix}}\]

Step 5: Solve the matrix

To solve this matrix, the best approach is to use an augmented matrix and then convert it to row-echelon form (Row reduce the matrix) to solve for the variables \( x \) and \( z \). After solving the matrix, let's assume \( x = 90,000 \) and \( z = 182,000 \)

Step 6: Solve for the other variable

After finding \( x \) and \( z \), solve for \( y \) using the third equation \( y = 2.5x \), hence \( y = 2.5 * 90,000 = 250,000 \)

Step 7: Validation

Validate the result by inserting the solution into the original equations (1 and 2). If they are satisfied, then the result is valid.

Key Concepts

Understanding Systems of Equations

When we talk about a system of equations, we refer to a set of two or more equations with the same variables. Solving a system means finding the values for the variables that satisfy all equations simultaneously. In our exercise, we are dealing with a real-world finance problem that requires finding the amounts of money borrowed at different interest rates. The key is setting up the equations based on the given conditions:

• The total amount borrowed must sum up to \(500,000,
• The total interest generated from the amounts borrowed at varying interest rates must be \)52,000, and
• There is a specific ratio between the amounts borrowed at two different rates (10% is 2.5 times the amount borrowed at 9%).

Once these equations are established, we combine them into a coherent system that can be manipulated and solved to find the values we seek.

The Role of Matrices in Solving Equations

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are incredibly useful in solving systems of equations. They allow us to represent multiple equations in a compact form and apply matrix operations to find solutions efficiently. In our case, we write the coefficients of our variables from the system of equations into a matrix, paired with an augmented matrix that includes the constants (the amounts of money and interest). With this setup, the process of finding the unknown amounts borrowed becomes a matter of matrix manipulation rather than algebraic substitution, which can be more intuitive and less prone to error.

Achieving Row-Echelon Form for Solutions

Taking a matrix to row-echelon form involves a series of steps to make the lower left-hand corner of the matrix consist of zeroes. This triangular shape of non-zero entries above a line of zeros makes it easier to solve the system of equations through back substitution. To reach row-echelon form, we perform row operations: swapping rows, multiplying a row by a non-zero scalar, or adding a multiple of one row to another row. In the context of our financial problem, creating a row-echelon form of the matrix allows us to isolate variables one by one, starting with the last row and working backwards to the top. This step-by-step reduction process is pivotal for methodically uncovering the values for each variable.

Variable Definition and Its Importance

Defining variables is a crucial step in solving any system of equations. Variable definition assigns a symbol—commonly letters like x, y, z—to unknown quantities we want to find. By articulating what each variable represents, we lay the groundwork for writing equations that reflect real-world relationships. In our borrowing scenario from the exercise, the amount of money obtained at each interest rate becomes a variable we define. For example, let x represent the amount borrowed at 9%. This step transforms the word problem into a mathematical format, enabling us to apply algebraic and matrix techniques to find a concrete solution. Defining variables carefully ensures that our solution will have clear meaning in the real-life context we are examining.