Math Problem Statement

Solve the following equestions using iteration method

8x - y + 2z = 13 3x + 2y + 12z = 25 x - 10y + 3z = 17

Solution

The iterative method is often used for solving systems of linear equations, such as the Jacobi or Gauss-Seidel methods. Here, I will use the Jacobi method for clarity. Let's rewrite the equations to isolate one variable in each equation.

Given:

  1. 8xy+2z=138x - y + 2z = 13
  2. 3x+2y+12z=253x + 2y + 12z = 25
  3. x10y+3z=17x - 10y + 3z = 17

Step 1: Rearrange equations to isolate each variable:

  1. x=18(13+y2z)x = \frac{1}{8}(13 + y - 2z)
  2. y=12(253x12z)y = \frac{1}{2}(25 - 3x - 12z)
  3. z=13(17x+10y)z = \frac{1}{3}(17 - x + 10y)

Step 2: Initial guesses

We start with initial guesses: x0=0,y0=0,z0=0x_0 = 0, y_0 = 0, z_0 = 0.

Step 3: Iterations

We iterate using the formulas derived above.

Iteration 1:

  • x1=18(13+020)=1.625x_1 = \frac{1}{8}(13 + 0 - 2 \cdot 0) = 1.625
  • y1=12(2530120)=12.5y_1 = \frac{1}{2}(25 - 3 \cdot 0 - 12 \cdot 0) = 12.5
  • z1=13(170+100)=5.667z_1 = \frac{1}{3}(17 - 0 + 10 \cdot 0) = 5.667

Iteration 2:

  • x2=18(13+12.525.667)=1.104x_2 = \frac{1}{8}(13 + 12.5 - 2 \cdot 5.667) = 1.104
  • y2=12(2531.625125.667)=16.125y_2 = \frac{1}{2}(25 - 3 \cdot 1.625 - 12 \cdot 5.667) = -16.125
  • z2=13(171.625+1012.5)=43.625z_2 = \frac{1}{3}(17 - 1.625 + 10 \cdot 12.5) = 43.625

Iteration 3:

  • x3=18(1316.125243.625)=13.8125x_3 = \frac{1}{8}(13 - 16.125 - 2 \cdot 43.625) = -13.8125
  • y3=12(2531.1041243.625)=243.208y_3 = \frac{1}{2}(25 - 3 \cdot 1.104 - 12 \cdot 43.625) = -243.208
  • z3=13(171.104+1016.125)=137.042z_3 = \frac{1}{3}(17 - 1.104 + 10 \cdot -16.125) = -137.042

The process continues until the values converge (the difference between successive iterations becomes negligible).

Solution Criteria

Would you like me to compute more iterations for convergence or explore another iterative method (e.g., Gauss-Seidel)? Let me know!


Expanding Questions:

  1. Would you like an explanation of how convergence is achieved in iterative methods?
  2. Should I plot the convergence of variables to show the process visually?
  3. Do you want to explore the Gauss-Seidel method for comparison?
  4. Is there a specific tolerance level you'd like to set for stopping iterations?
  5. Do you want me to verify the solution once obtained?

Tip: When using iterative methods, always check for convergence criteria, such as the dominance of the diagonal in the matrix.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Iterative Methods
System of Linear Equations

Formulas

x = (1/8)(13 + y - 2z)
y = (1/2)(25 - 3x - 12z)
z = (1/3)(17 - x + 10y)

Theorems

Jacobi Method
Convergence Criteria for Iterative Methods

Suitable Grade Level

Undergraduate (Mathematics or Engineering)