Math Problem Statement
Solution
The problem involves using matrix algebra to determine how many products can be produced based on the processing rate of three machines (A, B, and C) and the number of hours available on each machine.
Problem Restatement:
You are given the matrix for the number of units of three products (A, B, C) processed per hour on three machines (P, Q, R):
10 & 15 & 19 \\ 13 & 11 & 20 \\ 16 & 18 & 14 \end{pmatrix}$$ The number of hours available for machines A, B, and C are 54, 46, and 48, respectively. ### Steps for Solution: 1. **Define the Variables:** Let the number of units produced for products P, Q, and R be represented as a column matrix: $$\text{Product Output} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$$ where $$x_1, x_2, x_3$$ represent the number of units produced for products P, Q, and R, respectively. 2. **Matrix Setup:** The system of equations based on the machine processing rates and available hours is given by: $$\begin{pmatrix} 10 & 15 & 19 \\ 13 & 11 & 20 \\ 16 & 18 & 14 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 54 \\ 46 \\ 48 \end{pmatrix}$$ 3. **Solve the System of Equations:** To determine the number of units produced, you need to solve this system of linear equations using matrix algebra. The solution can be obtained by finding the inverse of the coefficient matrix (if it exists) and multiplying it by the vector of available hours: $$\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \text{Matrix of Coefficients}^{-1} \times \begin{pmatrix} 54 \\ 46 \\ 48 \end{pmatrix}$$ Would you like me to calculate the result for you?Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Linear Equations
Matrix Inversion
Formulas
AX = B (where A is the coefficient matrix, X is the variable matrix, and B is the constants matrix)
X = A^{-1} * B (if A is invertible)
Theorems
Inverse Matrix Theorem
Cramer's Rule (alternative method)
Suitable Grade Level
Grades 10-12
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