# 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

### Comments for Solve Equation in Matrix format

 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 ``` ---------------------------------------------

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