Problem 56 - A Graphical Approach to College Algebra

Question

Solve each problem. A group of students bought 3 soft drinks and 2 boxes of popcorn at a movie for \(\$ 18.50 .\) A second group bought 4 soft drinks and 3 boxes of popcorn for \(\$ 26\). (a) Find a matrix equation \(A X=B\) whose solution gives the individual prices of a soft drink and a box of popcorn. Solve this matrix equation by using \(A^{-1}\) (b) Could these prices be determined if both groups had bought 3 soft drinks and 2 boxes of popcorn for \(\$ 18.50 ?\) Try to calculate \(A^{-1}\) and explain your results.

Step-by-Step Solution

Answer

The price of a soft drink is \( \$3.50 \) and a box of popcorn is \( \$4.00 \). Identical purchases can't determine prices due to lack of unique solutions.

Step 1: Define the Variables

Let the price of a soft drink be \( x \) and the price of a box of popcorn be \( y \). We will express the prices in terms of a system of equations using the given information.

Step 2: Set Up System of Equations

From the information given, we have the following system of equations:1. \( 3x + 2y = 18.50 \)2. \( 4x + 3y = 26 \)

Step 3: Express as Matrix Equation

We can express the system as a matrix equation \( A \mathbf{X} = B \) where:\[ A = \begin{bmatrix} 3 & 2 \ 4 & 3 \end{bmatrix}, \mathbf{X} = \begin{bmatrix} x \ y \end{bmatrix}, B = \begin{bmatrix} 18.5 \ 26 \end{bmatrix} \]

Step 4: Find the Inverse of Matrix A

First, find the determinant of \( A \): \[ \text{det}(A) = (3)(3) - (2)(4) = 9 - 8 = 1 \]Since \( \text{det}(A) eq 0 \), matrix \( A \) is invertible. The inverse of \( A \) is given by:\[ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} 3 & -2 \ -4 & 3 \end{bmatrix} = \begin{bmatrix} 3 & -2 \ -4 & 3 \end{bmatrix} \]

Step 5: Solve for X using Inverse

Multiply both sides of the equation \( A \mathbf{X} = B \) by \( A^{-1} \) to solve for \( \mathbf{X} \):\[ \mathbf{X} = A^{-1} B = \begin{bmatrix} 3 & -2 \ -4 & 3 \end{bmatrix} \begin{bmatrix} 18.5 \ 26 \end{bmatrix} \]Calculating the multiplication:\[ \mathbf{X} = \begin{bmatrix} 3(18.5) - 2(26) \ -4(18.5) + 3(26) \end{bmatrix} = \begin{bmatrix} 55.5 - 52 \ -74 + 78 \end{bmatrix} = \begin{bmatrix} 3.5 \ 4 \end{bmatrix} \]Thus, \( x = 3.5 \) and \( y = 4 \).

Step 6: Check the Second Scenario

If both groups bought identical quantities, the equations would be identical. In matrix form, \( A \) would become \( \begin{bmatrix} 3 & 2 \ 3 & 2 \end{bmatrix} \) which results in determinant zero: \( 3(2) - 2(3) = 0\). \( A \) is now singular, hence no inverse exists.

Key Concepts

Matrix Equation

A matrix equation is a mathematical expression where matrices are used to represent systems of equations. In this exercise, we're dealing with finding the prices of a soft drink and a box of popcorn by setting up and solving a matrix equation. The real-world challenge is expressed with the matrix equation form:

• Matrix \( A \), containing coefficients from the linear equations.
• Vector \( \mathbf{X} \), representing unknowns (soft drink and popcorn prices).
• Matrix \( B \), which contains the constants \( (18.50 \text{ and } 26) \).

To solve this, assume vector \( \mathbf{X} = \begin{bmatrix} x \ y \end{bmatrix} \), where \( x \) is the price of a soft drink and \( y \) is the price of popcorn. This setup allows translating the word problem directly into mathematical representation, facilitating a structured approach to solve real-world problems with algebra.

Inverse of a Matrix

The concept of an inverse of a matrix is crucial in solving matrix equations. An inverse matrix \( A^{-1} \) can only be found if the determinant of the matrix \( A \) is not zero. It is similar to how dividing by a number is possible if the number is not zero.

• To find \( A^{-1} \), calculate the determinant \( \text{det}(A) \).
• If \( \text{det}(A) = 1 eq 0 \), as in this exercise, the matrix is invertible.
• The inverse is derived using a specific formula applied to \( A \).

For our exercise, the inverse is particularly instrumental because it transforms a multi-dimensional problem into a solvable format by pre-multiplying both sides of the equation \( A \mathbf{X} = B \) with \( A^{-1} \), leading us directly to the solution \( \mathbf{X} = A^{-1} B \). This step simplifies carrying out calculations to determine the precise values for \( x \) and \( y \).

Systems of Linear Equations

Systems of linear equations consist of multiple linear equations that share common variables. They often appear in real-world problems where relationships between multiple factors are analyzed. In this case, the systems provide two equations stemming from the purchases of different student groups at the movies.

• The equations are: \( 3x + 2y = 18.50 \) and \( 4x + 3y = 26 \).
• Here, \( x \) and \( y \) represent the prices we wish to find.
• These can be solved using substitution, elimination, or, as shown here, matrix algebra.

With the help of matrices, especially when the equations are linearly independent (as indicated by a non-zero determinant), the solution can be found systematically. However, if the equations are not independent, as in part (b) of the exercise, where both groups bought identical items, the determinant becomes zero, indicating no unique solution can exist.