Math Problem Statement

10a+5b= -680.3938 4.5a+18b+4.5c=-957.906 4.5b+18c+4.5d=-957.906 4.5c+18d+4.5e=-957.906 4.5d+18e+4.5f=-957.906 4.5e+9f=-478.953

Find the unknown variables.

Solution

You have a system of 6 equations with 6 unknowns, which can be solved using substitution, elimination, or matrix methods. Let's summarize the system for clarity:

  1. 10a+5b=680.393810a + 5b = -680.3938
  2. 4.5a+18b+4.5c=957.9064.5a + 18b + 4.5c = -957.906
  3. 4.5b+18c+4.5d=957.9064.5b + 18c + 4.5d = -957.906
  4. 4.5c+18d+4.5e=957.9064.5c + 18d + 4.5e = -957.906
  5. 4.5d+18e+4.5f=957.9064.5d + 18e + 4.5f = -957.906
  6. 4.5e+9f=478.9534.5e + 9f = -478.953

The most efficient way to solve this system is to use matrix methods (Gaussian elimination or matrix inverses). I'll use a direct approach to solve for all unknown variables.

Let me calculate the solution for you. numpy as np

Coefficients matrix

A = np.array([ [10, 5, 0, 0, 0, 0], [4.5, 18, 4.5, 0, 0, 0], [0, 4.5, 18, 4.5, 0, 0], [0, 0, 4.5, 18, 4.5, 0], [0, 0, 0, 4.5, 18, 4.5], [0, 0, 0, 0, 4.5, 9] ])

Constants vector

B = np.array([-680.3938, -957.906, -957.906, -957.906, -957.906, -478.953])

Solving the system of equations

solution = np.linalg.solve(A, B) solutionThe solution to the system of equations is as follows:

  • a=52.5935a = -52.5935
  • b=30.8918b = -30.8918
  • c=36.7073c = -36.7073
  • d=35.1470d = -35.1470
  • e=35.5726e = -35.5726
  • f=35.4307f = -35.4307

These are the values of the unknown variables.

Would you like more details or an explanation of the solution process? Also, here are some related questions you might find interesting:

  1. How can you check the solution of a system of equations?
  2. What is Gaussian elimination and how does it relate to solving systems of linear equations?
  3. How do matrix inverses help in solving linear systems?
  4. Can this system be solved using substitution or elimination methods?
  5. What are the advantages of using numerical methods for solving systems of equations?

Tip: In practice, always verify the solution by substituting the values back into the original equations to confirm they satisfy all of them.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Methods

Formulas

Ax = B (Matrix equation)
Gaussian elimination
Matrix inverse method

Theorems

Cramer's Rule
Gaussian Elimination
Inverse Matrix Theorem

Suitable Grade Level

Grades 11-12