Problem 109 - Precalculus with Limits

Question

TIPS A food server examines the amount of money earned in tips after working an 8-hour shift. The server has a total of \(\$95\) in denominations of \(\$1\), \(\$5\), \(\$10\), and \(\$20\) bills. The total number of paper bills is 26. The number of \(\$5\) bills is 4 times the number of \(\$10\) bills, and the number of \(\$1\) bills is 1 less than twice the number of \(\$5\) bills. Write a system of linear equations to represent the situation. Then use matrices to find the number of each denomination.

Step-by-Step Solution

Answer

To obtain the actual numerical solution, you have to perform the mathematical operations to solve the matrix. We need to substitute the relationships among the variables into the equations to solve for the variables. For instance, replace \(y\) in terms of \(z\) and \(x\) in terms of \(y\) in the first two equations and solve for \(z\) and \(w\). Then, substitute these values back into the equations for \(y\) and \(x\). After solving the matrix, you will be able to determine the counts of each denomination.

Step 1: Define the Variables

Let's call the number of $1 bills as \(x\), the number of $5 bills as \(y\), the number of $10 bills as \(z\) and $20 bills as \(w\).

Step 2: Formulate the System of Equations

The problem presents four conditions that translate into four equations. Firstly, the total sum of the value of all bills is $95, which gives us the equation: \(x + 5y + 10z + 20w = 95\). Second, the total number of bills is 26, hence \(x + y + z + w = 26\). Third, the number of $5 bills is four times the number of $10 bills, giving us \(y=4z\). Lastly, the number of $1 bills is one less than twice the number of $5 bills, which translates into \(x=2y-1\).

Step 3: Represent the System of Equations as a Matrix

The matrix representation of this system of equations is as follows: \[ \left[ \begin{array}{cccc} 1 & 5 & 10 & 20 \ 1 & 1 & 1 & 1 \ 0 & 4 & -1 & 0 \ 2 & -1 & 0 & 0 \end{array} \right] * \left[ \begin{array}{c} x \ y \ z \ w \end{array} \right] = \left[ \begin{array}{c} 95 \ 26 \ 0 \ 1 \end{array} \right] \]

Step 4: Solve the Matrix

You can now use a technique such as Gaussian Elimination or Cramer's Rule to solve this system of equations.

Key Concepts

Matrices

Matrices are rectangular arrays of numbers or expressions arranged in rows and columns. They are a vital tool in linear algebra, enabling us to handle systems of linear equations efficiently. In the problem provided, we used a matrix to represent the system of linear equations that models the tip scenario. The matrix's rows correspond to each equation, and the columns match each variable, providing a structured way to handle multiple equations at once.

When you translate a system of equations to a matrix, you create two parts:

• The coefficient matrix, which contains only the coefficients of the variables.
• The constant matrix, which contains the constants from the right-hand side of the equations.

This representation allows us to apply systematic methods such as Gaussian Elimination to find the solution, as you will see later.

Linear Algebra

Linear Algebra is a branch of mathematics that concerns itself with vector spaces (also called linear spaces) and linear mappings between these spaces. It deals with concepts like matrices, vectors, determinants, and systems of linear equations, which are crucial in various fields including engineering, physics, computer science, and econometrics.

In our exercise, linear algebra helps us systematically tackle the four conditions given by:

• The total value of bills.
• Number of bills.
• Ratio of $5 to $10 bills.
• Relation among $1 and $5 bills.

These conditions form a system of linear equations, which can be represented and solved using linear algebra techniques, showing the practical applications of this field.

Gaussian Elimination

Gaussian Elimination is perhaps one of the most frequently used methods for solving systems of linear equations. It involves using row operations to transform a matrix into a row-echelon form, making it easier to find solutions.

During Gaussian Elimination, three types of row operations are used to simplify the matrix:

• Swapping two rows.
• Multiplying a row by a nonzero constant.
• Adding or subtracting a multiple of one row from another row.

The technique simplifies the system until it is straightforward to identify the solutions by back substitution. In our task, Gaussian Elimination is a suitable method to solve the matrix that represents the system of equations formed by the tip problem.

Cramer's Rule

Cramer's Rule is another method for solving a system of linear equations, but it applies specifically to systems where the coefficient matrix is square (i.e., it has the same number of rows as columns) and where the determinant is non-zero. This rule utilizes determinants to find the solution directly.

To use Cramer's Rule, you would:

• Calculate the determinant of the coefficient matrix.
• For each variable, replace its column in the matrix with the constant matrix and calculate the determinant of this new matrix.
• The solution for each variable is the ratio of the determinant of this new matrix to the determinant of the original coefficient matrix.

Although Cramer's Rule is not commonly used for large systems due to its inefficiency, it provides a straightforward, formulaic method for solving small systems, like 2x2 or 3x3, where applicable.