SOLUTION: Solve the following system of equations by Gauss-Elimination method
x + y + z = 9
x – 2y + 3z = 8
2x + y – z = 3
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Question 235843: Solve the following system of equations by Gauss-Elimination method
x + y + z = 9
x – 2y + 3z = 8
2x + y – z = 3
Answer by Stitch(470) (Show Source): You can put this solution on YOUR website!
Aghhh so that was not Gauss-Elimination but below is a nother way to do it. Sorry about that....
Here is the Gauss-Elimination way:
Equation 1:
Equation 2:
Equation 3:
-*(Equation 1) + Equation 2 = Equation 2
Multiply equation 1 by negative 1 and then add to equation 2
-(X + Y + Z = 9) + (X - 2Y +3Z = 8) Rewrite the equation
(-X - Y - Z = -9) + (X - 2Y +3Z = 8) Combine like terms, notice that the X's cancel out
Equation 2 now equals
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To eliminate X for equation 3, multiply equation 1 by -2
-2*(Equation 1) + Equation 3 = Equation 3
-2*(X + Y + Z = 9) + (2X + Y - Z = 3) Rewrite the equation
(-2X - 2Y -2Z = -18) + (2X + Y - Z = 3) Combine like terms, notice that the X's cancel out
Equation 3 now equals
-------
Now we have to eliminate another variable
Equation 2 now equals
Equation 3 now equals
-3*(Equation 3) + Equation 2 = Equation 2
-3*(-Y -3Z = -15) + (-3Y + 2Z = -1) Rewrite the equation
(3Y + 9Z = 45) + (-3Y + 2Z = -1) Combine like terms, notice that the Y's cancel out
Equation 2 now equals Divide both sides by 11
Now we have to go back and do substitution
Lets plug 4 in for Z in the new equation 3
Simplify
Add Y to both sides
Add 15 to both sides
Now lets plug our answers into equation 1
Combione like terms
Subtract 7 from both sides
You can see the checks down below
-------------------------------------------------------------------------------
Anothor way
Given:
Equation 1:
Equation 2:
Equation 3:
------------------------------------
Solution:
The first step is to solve for 1 of the variables.
Lets set equations 1 & 3 equal to zero
Equation 1: Subtract 9 from both sides
Equation 3: Subtract 3 from both sides
Now since both equations equal zero, we can set them equal to each other
Add 3 to both sides
Subtract X from both sides
Combine like terms
Subtract Y from both sides
Add Z to both sides
Combine like terms
Lets label as equation 4
-------------------------------------------
Now plug (2Z - 6) into equation 1 for X
Equation 1:
Combine like terms
Add 6 to both sides
Subtract 3Z from both sides
Lets label as equation 5
-------------------------------------------
Now plug equation 4 in for X, and equation 5 in for Y in Equation 2
Equation 2:
Combine like terms
Add 36 to both sides
Divide both sides by 11
-------------------------------------------
Now plug 4 into equation 5 for Z
Equation 5:
Simplify
Combine like terms
-------------------------------------------
Now plug 4 into equation 4 for Z
Equation 4:
Simplify
Combine like terms
-------------------------------------------
Its time to check your answers
X = 2, Y = 3, Z = 4
Plug your answers into the three given equations and make sure that they are true
Equation 1:
Equation 2:
Equation 3:
----
Equation 1:
Equation 1 checks out
----
Equation 2:
Equation 2 checks out
----
Equation 3:
Equation 3 checks out
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