Problem 110 - Precalculus with Limits

Question

BANKING A bank teller is counting the total amount of money in a cash drawer at the end of a shift. There is a total of \(\$2600\) in denominations of \(\$1\), \(\$5\), \(\$10\), and \(\$20\) bills The total number of paper bills is 235. The number of \(\$20\) bills is twice the number of \(\$1\) bills, and the number of \(\$5\) bills is 10 more than the number of \(\$1\) bills. Write a system of linear equations to represent the situation. Then use matrices to find then umber of each denomination.

Step-by-Step Solution

Answer

Solve the value of each variable using matrix multiplication. The four variables in X = [\(X_1\), \(X_5\), \(X_{10}\), \(X_{20}\)] represent the number of each bill denomination ($1, $5, $10, $20 respectively).

Step 1: Setting up the equations

We can interpret the given conditions into the following system of equations:1. \(X_1 + X_5 + X_{10} + X_{20} = 235\) because the total number of bills is 235. 2. \(1*X_1 + 5*X_5 + 10*X_{10} + 20*X_{20} = 2600\) because the total value of all bills is $2600.3. \(X_{20} = 2*X_1\) because the number of $20 bills is twice the number of $1 bills.4. \(X_5 = X_1 + 10\) because the number of $5 bills is 10 more than the number of $1 bills.

Step 2: Transforming the system into matrix form

Let’s convert our system to matrix form \(AX = B\), where A is a matrix of coefficients, B is the column of answers, and X is the column containing our variables [\(X_1, X_5, X_{10}, X_{20}\)]. The matrices will be:\(A = [[1, 1, 1, 1], [1, 5, 10, 20], [-2, 0, 0, 1], [-1, 1, 0, 0]]\)\(B = [235, 2600, 0, 10]\)\(X = [X_1, X_5, X_{10}, X_{20}]\)

Step 3: Solving the matrix equation

To find X, the matrix equation will be:\(X = A^{-1}B\)Where \(A^{-1}\) is the inverse of the matrix A. Compute the inverse of A, perform the multiplication with B, and use these results to find the value for each variable.

Key Concepts

Matrix Algebra

Matrix algebra is a handy tool in solving systems of linear equations, especially when dealing with multiple variables. It helps us organize complex sets of equations in a more structured form. This is done using matrices, which are rectangular arrays of numbers arranged in rows and columns. For example, in the exercise provided, we had several equations to represent different amounts and types of bills. By leveraging matrix algebra:

• We transformed these equations into a matrix form, generating a coefficient matrix, a variable matrix, and a result matrix.
• This method simplifies complex arithmetic operations, making calculations more manageable.

Each row of the matrix represents a linear equation, and each column corresponds to the coefficients of the variables in these equations. This way, matrix algebra provides a systematic approach to work with linear equations without confusing the variables or equations.

Inverse Matrix

An inverse matrix, often denoted as \( A^{-1} \), is like the reciprocal of a number but in matrix form. When it exists, multiplying a matrix by its inverse yields the identity matrix, much like how a number multiplied by its reciprocal equals one. In the context of solving linear equations:

• Finding an inverse matrix helps to solve equations through multiplication, effectively isolating the variable matrix.
• The inverse matrix concept is crucial because sometimes directly finding the solution might not be feasible due to complexity.
• We use the inverse to manipulate our equation system into a simpler form.
• In our exercise, we found the inverse of the coefficient matrix \( A \) to solve for the variable matrix \( X \). This was done by multiplying the inverse matrix by the result matrix \( B \).

However, it's important to note that not all matrices have an inverse. A matrix must be square (same number of rows and columns) and its determinant should not be zero for the inverse to exist.

Problem Solving with Matrices

Problem-solving with matrices is an efficient method for handling systems of linear equations, such as those encountered in our bank teller scenario. By using matrices, you can deal with multiple equations simultaneously. The steps generally include:

• Writing down the system of equations and transforming them into matrix form.
• Determining if the inverse of the coefficient matrix exists, which allows us to solve the system using the matrix equation \( X = A^{-1}B \).
• Computationally finding the inverse of matrix \( A \) and then performing the matrix multiplication with \( B \) to find \( X \).

By organizing information into matrices, solutions become clearer and more structured. This method manages to simplify the seemingly convoluted nature of simultaneous equations, offering a straightforward path to the solution. It's particularly advantageous in economics, engineering, and computer science, where problems often comprise multiple variables needing simultaneous consideration.