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Solve Equation in Matrix format










































Gaussian elimination method

Use Gaussian elimination method to solve the following systems of linear equations

   • 2X₁ + 2X₂ + X₃ = 9

   • X₁ + X₃ = 4

   • 4X₂ - 3X₃ = 17

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Oct 06, 2012
Solve Equations using Gaussian Elimination
by: Staff


Answer:

Part I

The system of equations in your problem statement is:



2X1 + 2X2 + 1X3 = 9
1X1 + 0 + 1X3 = 4
0 + 4X2 - 3X3 = 17

The augmented matrix for this system of equations is:



+2 +2 +1 : +9
+1 +0 +1 : +4
+0 +4 -3 : +17

The row operations used by the Gaussian elimination method are:



+2 +2 +1 : +9
+1 +0 +1 : +4
+0 +4 -3 : +17


          Divide row1 by 2



+1 +1 +1/2 : +9/2
+1 +0 +1 : +4
+0 +4 -3 : +17

          Add (-1 * row1) to row2



+1 +1 +1/2 : +9/2
+0 -1 +1/2 : -1/2
+0 +4 -3 : +17


          Divide row2 by -1



+1 +1 +1/2 : +9/2
+0 +1 -1/2 : +1/2
+0 +4 -3 : +17


          Add (-4 * row2) to row3



+1 +1 +1/2 : +9/2
+0 +1 -1/2 : +1/2
+0 +0 -1 : +15


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Oct 06, 2012
Solve Equations using Gaussian Elimination
by: Staff


---------------------------------------------

Part II

          Divide row3 by -1



+1 +1 +1/2 : +9/2
+0 +1 -1/2 : +1/2
+0 +0 +1 : -15


          Add (1/2 * row3) to row2



+1 +1 +1/2 : +9/2
+0 +1 +0 : -7
+0 +0 +1 : -15

          Add (-1/2 * row3) to row1



+1 +1 +0 : +12
+0 +1 +0 : -7
+0 +0 +1 : -15

          Add (-1 * row2) to row1



+1 +0 +0 : +19
+0 +1 +0 : -7
+0 +0 +1 : -15



Convert the final matrix back into equation form:



1X1 + 0X2 + 0X3 = +19
0X1 + 1X2 + 0X3 = -7
0X1 + 0X2 + 1X3 = -15



---------------------------------------------

Oct 06, 2012
Solve Equations using Gaussian Elimination
by: Staff


---------------------------------------------

Part III

Final Answer:


                 X₁ = 19

                 X₂ = -7

                 X₃ = -15

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Check the answer by substituting the numerical values of X₁, X₂ and X₃ into the original equations:

2X₁ + 2X₂ + X₃ = 9

2*19 + 2*(-7) + (-15) = 9

38 - 14 - 15 = 9

9 = 9, correct


X₁ + X₃ = 4

19 + (-15) = 4

4 = 4, correct



4X₂ - 3X₃ = 17

4*(-7) - 3(-15) = 17

-28 + 45 = 17

17 = 17, correct


Since the numerical values of X₁, X₂, and X₃ work in all three of the original equations, the solutions are correct.



Thanks for writing.

Staff
www.solving-math-problems.com


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