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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-> 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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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
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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
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Anothor way
Given:
Equation 1:
Equation 2:
Equation 3:
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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
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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
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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
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Now plug 4 into equation 5 for Z
Equation 5: Simplify Combine like terms
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Now plug 4 into equation 4 for Z
Equation 4: Simplify Combine like terms
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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:
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Equation 1: Equation 1 checks out
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Equation 2: Equation 2 checks out
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Equation 3: Equation 3 checks out
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