Math Problem Statement

Amila, Babu and Christy earn a total of Rs. 970 per week as the take home salary. Amila's take home salary plus twice each of the other's is Rs. 1640. Christy's take-home salary plus twice each of the other's is Rs. 1600. Using the matrices, find the take home salary of each.

Solution

Let the take-home salaries of Amila, Babu, and Christy be denoted by AA, BB, and CC respectively. We are given the following system of equations based on the problem description:

  1. Total salary equation:
    A+B+C=970A + B + C = 970

  2. Amila's salary equation:
    A+2B+2C=1640A + 2B + 2C = 1640

  3. Christy's salary equation:
    C+2A+2B=1600C + 2A + 2B = 1600

We can represent these equations in matrix form as:

1 & 1 & 1 \\ 1 & 2 & 2 \\ 2 & 2 & 1 \end{pmatrix} \begin{pmatrix} A \\ B \\ C \end{pmatrix} = \begin{pmatrix} 970 \\ 1640 \\ 1600 \end{pmatrix}$$ Let me solve this system of linear equations using matrix methods.The take-home salaries of each individual are as follows: - Amila's salary: Rs. 300 - Babu's salary: Rs. 330 - Christy's salary: Rs. 340 Would you like a detailed explanation of how this solution was reached, or do you have any questions? Here are 5 related questions you might consider: 1. How do you solve a system of linear equations using matrices? 2. What are other methods for solving systems of equations besides matrices? 3. How do you calculate the inverse of a matrix? 4. Can systems of linear equations always be solved using matrices? 5. What happens if the system of equations is inconsistent or dependent? **Tip:** Always verify the consistency of your system of equations before solving, as some systems may not have a unique solution.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
System of Linear Equations
Matrices

Formulas

Matrix representation of a system of linear equations
Solving system of equations using matrix methods (e.g., Gaussian elimination, inverse matrix)

Theorems

Matrix Inversion Theorem
Linear Combination of Equations

Suitable Grade Level

Grade 10-12