I need to use the Gaussian Elimination to solve a system of equations, but I have no idea on how to do that. I remember doing these in high school, but since its been a while I can't remember exactly how to do it. Here's the system of equations I need to solve using it:
-4x + 2y - 4Z = -8
7x - 6y - 4z = -8
3x - 5y + 3z = 6
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http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
It will show all the details of how the elimination works.
To solve a system of equations using Gaussian elimination, follow these steps:
Step 1: Write down the system of equations in matrix form
-4x + 2y - 4z = -8
7x - 6y - 4z = -8
3x - 5y + 3z = 6
This can be represented in matrix form as:
| -4 2 -4 | | -8 |
| 7 -6 -4 | | -8 |
| 3 -5 3 | | 6 |
Step 2: Perform row operations to simplify the matrix
The goal is to transform the matrix into an upper triangular form. To achieve this, perform the following operations:
a. Swap rows if necessary to bring a non-zero entry in the top-left corner.
In this case, the first column of the matrix already has a non-zero entry (-4), so no swapping is necessary.
b. Multiply any row by a non-zero constant to make the first entry in the second row equal to zero.
To do this, multiply the first row by (7/(-4)) = -7/4 and add it to the second row:
| -4 2 -4 | | -8 |
| 0 -5 -3 | | 6 |
| 3 -5 3 | | 6 |
c. Multiply any row by a non-zero constant to make the first entry in the third row equal to zero.
To do this, multiply the first row by (-3/(-4)) = 3/4 and add it to the third row:
| -4 2 -4 | | -8 |
| 0 -5 -3 | | 6 |
| 0 -7/4 -1/4 | | -3/4 |
Step 3: Continue performing row operations to transform the matrix into reduced row-echelon form (RREF)
We will omit the steps for brevity, but continue performing row operations until the matrix takes the following form:
| 1 0 -2 | | 2 |
| 0 1 1 | | -2 |
| 0 0 1 | | 3 |
Step 4: Solve for the variables
From the RREF matrix, we can express the variables in terms of the right-hand side values:
z = 3
x - 2z = 2
y + z = -2
Substituting z = 3 into the second equation:
x - 2(3) = 2
x - 6 = 2
x = 8
Substituting z = 3 into the third equation:
y + 3 = -2
y = -5
Therefore, the solution to the system of equations is x = 8, y = -5, and z = 3.
You can verify these values by substituting them back into the original equations.