Problem 60 - Precalculus Mathematics for Calculus

Question

A roadside fruit stand sells apples at \(75 \notin\) a pound, peaches at \(90 \notin\) a pound, and pears at \(60 \notin\) a pound. Muriel buys 18 pounds of fruit at a total cost of 13.80 dollar Her peaches and pears together cost 1.80 dollar 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 pounds of apples, 8 pounds of peaches, and 4 pounds of pears.

Step 1: Define the Variables

Let's define the variables for the fruit weights Muriel bought:- Let \( x \) be the pounds of apples.- Let \( y \) be the pounds of peaches.- Let \( z \) be the pounds of pears.

Step 2: Set Up the First Equation

The first equation comes from the total pounds of fruit bought, which is given as 18 pounds. Therefore, we have:\[ x + y + z = 18 \]

Step 3: Set Up the Second Equation

The second equation relates to the total cost being \$13.80. Considering the costs:- Apples cost \(0.75\) per pound,- Peaches cost \(0.90\) per pound,- Pears cost \(0.60\) per pound.Thus, the equation is:\[ 0.75x + 0.90y + 0.60z = 13.80 \]

Step 4: Set Up the Third Equation

The third equation comes from the condition that the cost of peaches and pears is \$1.80 more than the cost of apples:\[ 0.90y + 0.60z = 0.75x + 1.80 \]Simplifying, we get:\[ 0.90y + 0.60z - 0.75x = 1.80 \]

Step 5: Write the System of Equations

The system of equations based on the above conditions is:1. \( x + y + z = 18 \)2. \( 0.75x + 0.90y + 0.60z = 13.80 \)3. \( -0.75x + 0.90y + 0.60z = 1.80 \)

Step 6: Form the Coefficient Matrix and Constants Vector

The coefficient matrix \( A \) and the constants vector \( B \) from the system are:\[ A = \begin{bmatrix} 1 & 1 & 1 \ 0.75 & 0.90 & 0.60 \ -0.75 & 0.90 & 0.60 \end{bmatrix} \] \[ B = \begin{bmatrix} 18 \ 13.80 \ 1.80 \end{bmatrix} \]

Step 7: Calculate Determinants for Cramer's Rule

First, find the determinant of \( A \), \( \Delta \):\[ \Delta = \begin{vmatrix} 1 & 1 & 1 \ 0.75 & 0.90 & 0.60 \ -0.75 & 0.90 & 0.60 \end{vmatrix} \]Calculate it, using the formula for determinants of a 3x3 matrix.

Step 8: Solve for Determinants Using Cramer's Rule

Find determinants \( \Delta_x \), \( \Delta_y \), and \( \Delta_z \) by replacing the respective columns of \( A \) with \( B \) and solve. For example, for \( \Delta_x \):\[ \Delta_x = \begin{vmatrix} 18 & 1 & 1 \ 13.80 & 0.90 & 0.60 \ 1.80 & 0.90 & 0.60 \end{vmatrix} \]Solve similarly for \( \Delta_y \) and \( \Delta_z \).

Step 9: Compute the Variables

Using Cramer's Rule:\[ x = \frac{\Delta_x}{\Delta} \]\[ y = \frac{\Delta_y}{\Delta} \]\[ z = \frac{\Delta_z}{\Delta} \]

Step 10: Present Solution Numbers

After calculations, the solution is:- \( x = 6 \) pounds of apples- \( y = 8 \) pounds of peaches- \( z = 4 \) pounds of pears

Key Concepts

Linear Systems

In mathematics, a linear system is a collection of linear equations involving the same set of variables. These systems can be solved using various methods, but one popular method is Cramer's Rule, especially when the system has as many equations as variables and the determinant of the matrix is non-zero. In the given exercise, Muriel's fruit purchase forms a linear system as it is based on three linear equations. Each equation represents different conditions of the purchase—total weight, total cost, and the cost relation between different fruits. When setting up a linear system, it's essential to define variables effectively to represent unknowns. For example:

• Let \( x \)- be the pounds of apples.
• Let \( y \)- be the pounds of peaches.
• Let \( z \)- be the pounds of pears.

These definitions help in transforming word problems into manageable mathematical equations. With these definitions, the system of equations given in the exercise is:

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

Each equation corresponds to a logical statement in the problem—total weight, cost, and cost relation. Solving the system reveals the exact amount of each fruit Muriel bought.

Determinants

Determinants are special numbers calculated from square matrices. They play a vital role in matrix algebra and solving systems of linear equations, particularly using Cramer's Rule. When solving our system of equations with Cramer's Rule, calculating determinants helps determine the unique solution for each variable.The determinant of a 3x3 matrix \( A \), like in Muriel's scenario, is found using the formula: \[\Delta = \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei − fh) − b(di − fg) + c(dh − eg) \]For the exercise, we determine \( \Delta \) from the coefficient matrix: \[A = \begin{bmatrix} 1 & 1 & 1 \0.75 & 0.90 & 0.60 \-0.75 & 0.90 & 0.60 \end{bmatrix}\]Calculating this determinant is crucial since it's used to check if a system has a unique solution (non-zero \( \Delta \)) and to find specific determinants \( \Delta_x \), \( \Delta_y \), and \( \Delta_z \).Each \( \Delta_i \) is calculated similarly by substituting the corresponding column with the constants from the system to find each variable using Cramer's Rule, ensuring comprehensive solutions to Muriel’s problem.

Matrix Algebra

Matrix algebra involves operations on matrices and is a strong tool for solving systems of linear equations like those in the exercise. It forms the basis for applying Cramer's Rule efficiently.A matrix is a rectangular arrangement of numbers into rows and columns. For solving a linear system, we create a coefficient matrix and a constants vector:

• Coefficient matrix \( A \) represents the linear equations.
• Constants vector \( B \) accounts for the solutions or total conditions.

In our example:\[A = \begin{bmatrix} 1 & 1 & 1 \0.75 & 0.90 & 0.60 \-0.75 & 0.90 & 0.60 \end{bmatrix}, \quad B = \begin{bmatrix} 18 \ 13.80 \ 1.80 \end{bmatrix}\]The matrix equations can then be succinctly represented as \( AX = B \), where \( X \) holds the variables (\( x, y, z \)). Specific operations such as finding the inverse or using determinants enable solving these equations. Through matrix algebra, variables are derived by relating changes in one matrix to corresponding changes in another, strategically and methodically uncovering solutions like Muriel's fruit purchases.