The Gaussian elimination Method of solving linear equations is one of the most efficient options that mathematicians can use. This method is so useful that it can help compute systems of linear equations of three variables that may not be easy to compute otherwise.
Carl Gauss Friedrich first introduced this elimination method. It is the algorithm computers use to determine the values of three variable equations. For PCs and work systems, this algorithm can help commute thousands of equations and, as such, is one of the most helpful solutions that mathematicians and statisticians should know.
The Gaussian elimination depends highly on matrices. Generally, one of the major use of matrix is to represent linear systems of equations. The Gaussian elimination fully works on the matrix representation, and it is from there, that simplifications are carried out systematically.
Apart from the fact that the Gaussian Elimination method can be used to solve linear equations, it can also be used to compute matrix ranks, determinants of squared matrixes, and the inverse of invertible matrices. However, due to the scope of this article, the focus would be on how to use this method to solve linear systems of equations with at least three variables.
Manually solving GEM is possible. However, it can be extremely time exhausting and confounding when manually used on complex variables and many equations. In that case, the best option to go for is a workstation.
The Row Echelon Matrix and The Gaussian Elimination Method
To find the solutions of linear equations using the GEM, all the coefficients of the equation will be arranged into a matrix. The aim would then be to obtain the row reduced form of the matrix.
Reducing a matrix simply means manipulating it to become an upper triangular matrix. An upper triangular matrix is where the lower half of a matrix is all zeros. An upper triangular matrix is also known as a row echelon form. If the triangular matrix leading coefficients are all 1s, the matrix is a reduced row echelon matrix. A reduced echelon matrix is called the Gauss-Jordan Elimination method.
In certain instances, the reduced or unreduced echelon form of matrices may be the most applicable for the GEM.
So summarily, we aim to achieve with the GEM to reduce the entries as much as possible, with the bottommost row having the highest zero entries.
The Gaussian Elimination Formula
The Gaussian Elimination is a Matrix-dependent rows manipulation that can be used to solve complex systems of linear equations with a large number of unknowns.
For example, let there be a matrix given as
with all the alphabets representing different numbers, then being able to manipulate the matrix rows such that it becomes any of these;
It means that the Gaussian elimination has being achieved.
The matrix identified as (1) is an upper triangular matrix and a row echelon matrix because the lower part of its left side is all zeros. The other matrix identified as (2) is the reduced row echelon matrix because all of its bottom-left sides are zero, and its leading coefficients are all 1s.
Identifying the bottom sides of a matrix and its Leading Coefficients
Also, as can be seen from the same two matrices above, the line equally separating both matrices actually show us the bottom left sides more clearly. Also, the numbers which the line cuts across are the leading coefficients. This is because there are non-zero numbers before them in their respective rows.
It is important to know that it is possible for a matrix to actually have all its entries as zeros following operations on it.
How to Calculate the Gaussian Elimination for System of Linear Equations?
The steps for using the GEM are as follows;
Step 1: Arrange all the coefficients of the linear equations in a matrix format
Step 2: Ensure all the coefficients of the unknown variables at the left side of the ‘=’ sign are on one part of the matrices, making the most significant of the whole. The coefficients of the other part after the ‘=’ are usually at the extreme right-hand side of the matrix.
Step 3: Consistently multiply the rows in the matrices with non-zero numbers that could be negative or positive.
Step 4: Continue the rows operations until a row echelon matrix or reduced echelon row matrix is reached.
Step 5: Starts the backward substitution and solve for each variable at a time
The Gaussian Elimination with Examples
Example 1
Determine the variables for the system of linear equations
-3x + 2y – z = -1
6x – 6y + 7z = -7
3x – 4y + 4z = -6
Solution
Step 1: Arrange all the coefficients of the linear equations in a matrix format
-3x + 2y – z = -1
6x – 6y + 7z = -7
3x – 4y + 4z = -6
Would be
As can be seen from the above matrix, the first three columns from the left side are the coefficient of the unknown variables before the ‘=’ sign. Also, the coefficients on the fourth column are for those at the other part of the ‘=’ sign. This is step 2.
Another important thing is the dotted line that separates the matrix equations. It is simply meant to show that it is the left side coefficients that we aim to achieve a row echelon matrix for. So we aim to have something like this
Step 3 and Step 4: Consistently multiply the rows in the matrices with non-zero numbers that could be negative or positive. Continue the rows operations until a row echelon matrix or reduced echelon row matrix is reached.
So the next step will be to make the first 6 in the second role zero. So we will multiply the first row by 2 and add all of the second-row units from each of the newly derived units of the first row. So we will have something like this below
So we have been able to get one of our zeros. However, it is important to note that it is only row 2 that will change. All the initials values of row 1 and row 3 will remain valid. This will always be the situation after every single operation. The row where a digit has to be zero will be the only one that will change.
This, therefore, means that our real matrix after getting one of our zeros will be
As can be seen from the above, only the second row has been changed.
Now we still have to change the number ‘3’ in the first column and third row to continue our goal for an upper triangular matrix.
Now, we can change that ‘3’ to zero by simply adding the first and third-row digits together. That will give us
Now the next step will be to change the ‘-2’ to zero. To do this we will multiply the second row by -1 and add it to the third row
-1 x 1st row digits = (-2 x -1, 5 x -1, -9 x -1) = (2, -5, 9)
Adding the new digits to the 3rd row digits will give (2 + -2, -5 + 3, 9 + -7) = (0, -2, 2)
As such the new matrix will be
Step 4: Continue the rows operations until a row echelon matrix or reduced echelon row matrix is reached.
From the above matrix, we have been able to achieve the row echelon matrix, which is the fourth step. A deeper look will see us with a triangular matrix below.
Note that this is not a reduced echelon matrix as the leading coefficient are not all 1s
Step 5: Starts the backward substitution and solve for each variable at a time
Arranging the above matrix into a linear solution, we will have
-3x + 2y –z = -1 …… (1)
-2y + 5z = – 9 ……… (2)
-2z = 2 …………………. (3)
For Eqn (3),
For eqn (2),
-2y + 5z = – 9
Since z =1,
-2y + 5(-1) = – 9
-2y = – 9 + 5 = -4
-2y = – 4
Y = 2
For eqn (1),
-3x + 2y –z = -1
Since z = 1, y =2, we have
-3x + 2 x 2 – -1 = -1
-3x + 4 + 1 = -1
-3x = -1 – 5
this means that the unknown variables are;
x =2, y = 2, and z = -1
Conclusion
This article has systematically explained all that it takes to solve the system of Linear equations using the Gaussian Elimination Method. This method is extremely trustworthy and a very reliable method to adopt for solving equations. All steps were systematically outlined to make it as easy as possible for students to understand. Therefore, while it is possible that you may get to solve other equations, sticking with these steps will help you get the best results.
