How do you solve using gaussian elimination or gauss-jordan elimination, #y + 3z = 6#, #x + 2y + 4z = 9#, #2x + y + 6z = 11#?
Using Gaussian elimination
Note that at this point rows [4] and [5] are in Gaussian form and all that remains is to zero out the first two columns of row [3]
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To solve the system of linear equations using Gaussian elimination or Gauss-Jordan elimination, we represent the equations in augmented matrix form and then perform row operations to transform the matrix into row-echelon form or reduced row-echelon form.
The given system of equations is:
[ \begin{align*} y + 3z &= 6 \ x + 2y + 4z &= 9 \ 2x + y + 6z &= 11 \ \end{align*} ]
Writing this system in augmented matrix form:
[ \left[ \begin{array}{ccc|c} 0 & 1 & 3 & 6 \ 1 & 2 & 4 & 9 \ 2 & 1 & 6 & 11 \ \end{array} \right] ]
Now, let's perform row operations to transform this matrix into row-echelon form:
- Multiply the first row by -1 and add it to the second row.
- Multiply the first row by -2 and add it to the third row.
This gives us:
[ \left[ \begin{array}{ccc|c} 0 & 1 & 3 & 6 \ 1 & 2 & 4 & 9 \ 0 & -1 & 0 & -1 \ \end{array} \right] ]
Now, continue with row operations to simplify the matrix:
- Multiply the second row by -1 and add it to the first row.
This gives:
[ \left[ \begin{array}{ccc|c} 1 & 0 & -1 & -3 \ 1 & 2 & 4 & 9 \ 0 & -1 & 0 & -1 \ \end{array} \right] ]
- Multiply the second row by 1 and add it to the third row.
This gives:
[ \left[ \begin{array}{ccc|c} 1 & 0 & -1 & -3 \ 0 & 1 & 4 & 6 \ 0 & -1 & 0 & -1 \ \end{array} \right] ]
Now, perform additional row operations to transform the matrix into reduced row-echelon form if desired. However, this matrix already provides enough information to find the solution.
From the matrix, we can see that:
[ \begin{align*} x &= -3 \ y &= 6 \ z &= 1 \ \end{align*} ]
So, the solution to the system of equations is ( x = -3 ), ( y = 6 ), and ( z = 1 ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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