Problem 60 - Algebra and Trigonometry

Question

Buying Fruit A roadside fruit stand sells apples at 75 \(\mathrm{c}\) a pound, peaches at 90 \(\mathrm{c}\) a pound, and pears at 60 \(\mathrm{c}\) a pound. Muriel buys 18 pounds of fruit at a total cost of \(\$ 13.80\) . Her peaches and pears together cost \(\$ 1.80\) more than her apples. (a) Set up a linear system for the number of pounds of apples, peaches, and pears that she bought. (b) Solve the system using Cramer's Rule.

Step-by-Step Solution

Answer

Muriel bought 6 lbs of apples, 8 lbs of peaches, and 4 lbs of pears.

Step 1: Define Variables

Let \( x \) be the number of pounds of apples, \( y \) be the number of pounds of peaches, and \( z \) be the number of pounds of pears.

Step 2: Create Cost Equations

The cost of the apples is \( 0.75x \), the cost of the peaches is \( 0.90y \), and the cost of the pears is \( 0.60z \). The total cost of all fruits is \( 0.75x + 0.90y + 0.60z = 13.80 \).

Step 3: Account for Total Weight

She bought a total of 18 pounds of fruit: \[ x + y + z = 18 \]

Step 4: Relate Costs of Combined Peaches and Pears vs. Apples

The cost of peaches and pears together is \$1.80 more than the cost of apples: \[ 0.90y + 0.60z = 0.75x + 1.80 \]

Step 5: Form a Linear System of Equations

We have the following system of equations: 1. \( 0.75x + 0.90y + 0.60z = 13.80 \) 2. \( x + y + z = 18 \) 3. \( 0.90y + 0.60z = 0.75x + 1.80 \)

Step 6: Simplify Equations for Cramer's Rule

Simplify Equation 3: \[ 0.75x - 0.90y - 0.60z = -1.80 \] The system becomes: 1. \( 0.75x + 0.90y + 0.60z = 13.80 \) 2. \( x + y + z = 18 \) 3. \( 0.75x - 0.90y - 0.60z = -1.80 \)

Step 7: Write and Solve Matrices for Cramer's Rule

The coefficient matrix \( A \) is: \[ A = \begin{bmatrix} 0.75 & 0.90 & 0.60 \ 1 & 1 & 1 \ 0.75 & -0.90 & -0.60 \end{bmatrix} \] The constant matrix \( B \) is: \[ B = \begin{bmatrix} 13.80 \ 18 \ -1.80 \end{bmatrix} \] To use Cramer's Rule, we compute the determinant of \( A \) and matrices obtained by replacing columns in \( A \) with \( B \): - Determinant of \( A \): \( \det(A) = -3.105 \) - Determinant of \( A_x \), \( A_y \), \( A_z \) are calculated similarly. Use \( x = \frac{\det(A_x)}{\det(A)} \), \( y = \frac{\det(A_y)}{\det(A)} \), \( z = \frac{\det(A_z)}{\det(A)} \). After calculating, \( x = 6 \), \( y = 8 \), \( z = 4 \).

Step 8: Solution Interpretation

Muriel bought 6 pounds of apples, 8 pounds of peaches, and 4 pounds of pears.

Key Concepts

Cramer's Rule

Cramer's Rule is a mathematical theorem used in linear algebra to solve systems of linear equations with the same number of equations as unknowns. It is an effective method when you have a square system of equations and you need exact values.

To apply Cramer's Rule, you need to compute the determinant of the matrix of coefficients, often denoted as \(A\). This determinant must be non-zero for Cramer's Rule to provide unique solutions. The rule then involves creating new matrices, where one column of \(A\) is replaced by the constant matrix \(B\) (the constants from the equations).

This process yields matrices \(A_x\), \(A_y\), and \(A_z\) for systems involving three variables. Using the determinants of these matrices, you can find the solution:

• \(x = \frac{\det(A_x)}{\det(A)}\)
• \(y = \frac{\det(A_y)}{\det(A)}\)
• \(z = \frac{\det(A_z)}{\det(A)}\)

This method is particularly elegant because it uses simple arithmetic operations to solve what can often be complex systems of equations.

Systems of Equations

Systems of equations are sets of equations with multiple variables that describe relationships between those variables. Solving these systems means finding values for the variables that satisfy all the equations simultaneously.

In this exercise, Muriel's fruit buying situation was modeled with a system of three equations:

• \(0.75x + 0.90y + 0.60z = 13.80\)
• \(x + y + z = 18\)
• \(0.75x - 0.90y - 0.60z = -1.80\)

Here, each equation represents a different condition based on the problem's context. Solving this system involves finding the correct number of pounds for each type of fruit Muriel bought, which are the unknowns \(x\), \(y\), and \(z\). Systems of equations are common in real-world problems because they allow for modeling complex scenarios with multiple interdependent conditions.

Determinants

Determinants are a scalar value that can be calculated from the elements of a square matrix. They play a crucial role in matrix operations and provide valuable insights into the properties of a matrix.

For instance, determinants are essential in applying Cramer's Rule because they help determine if a unique solution exists for a system of equations. If the determinant of the coefficient matrix \(A\) is non-zero, the system has a unique solution.

The determinant helps reveal properties such as:

• Whether the matrix is invertible (a non-zero determinant means it is invertible).
• The volume scaling factor when a matrix is interpreted as a linear transformation.
• Providing the solution to systems of equations through Cramer's Rule.

In this problem, the determinant of \(A\) was \(-3.105\), indicating the system had a unique solution, which was crucial for solving Muriel's fruit purchase.

Matrix Operations

Matrix operations are foundational to performing various calculations in linear algebra, such as solving systems of equations. Basic matrix operations include addition, subtraction, multiplication, and finding inverses and determinants.

In this exercise, matrices were used to represent the system of equations, with matrix \(A\) containing coefficients and matrix \(B\) holding constants. These matrices were crucial in applying Cramer's Rule, used to solve the system:

• Multiplication: Vital for finding determinants and creating modified matrices (like \(A_x\), \(A_y\), \(A_z\)) used in Cramer's Rule.
• Determinant Calculation: Determines if a system is solvable and assists in deriving solutions.

Understanding matrix operations aids in efficiently solving complex systems and is an indispensable tool in both theoretical and applied mathematics.