A system of linear equations can be solved by (1) preparing an Augmented Matrix, and then (2) using row operations to convert the Augmented Matrix into a triangular form. Once the Augmented Matrix has been converted to triangular form, the solution to every variable can often be read directly from the matrix with no further work.
This can be illustrated by solving for the variables x, y, and z in the following system of linear equations:
-4x - 3y - z = -12
-5x + 6y + 4z = -24
-3x - y - 6z = -22
First, arrange all three equations in standard form.
Ax + By + Cz = D
This has already been done in the statement of the problem:
-4x - 3y - z = -12
-5x + 6y + 4z = -24
-3x - y - 6z = -22
Prepare an Augmented Matrix of 3 rows and 4 columns.
Each row in the Augmented Matrix will represent one of the three equations.
The first number in each row is the coefficient for the x variable, the second number in each row is the coefficient for the y variable, the third number in each row is the coefficient for the z variable, and the last number will be the constant:
| -4 -3 -1:-12 |
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |
THIS IS the GOAL: Convert the Augmented Matrix into a triangular form. Once this is done, we are finished.
The triangular form will look like this: a diagonal pattern of 1's with 0's everywhere else but the last column.
| 1 0 0:? |
| 0 1 0:? |
| 0 0 1:? |
Once we have converted the matrix into the triangular form, the? shown in row 1 will be the solution value of x, the? shown in row 2 will be the solution value of y, and the? shown in row 3 will be the solution value of z.
To convert the augmented matrix into a triangular matrix, we will perform various row operations, one at a time.
To convert the convert the Augmented Matrix to a triangular form, the following operations can be used:
1. switch two of the rows
2. multiply any row by a number which is not zero
3. Replace any row with the result of adding that row to another row.
| -4 -3 -1:-12 |
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |
Note:
There are many calculations involved, which means there is a high probability of making errors in arithmetic. For this reason I recommend using a calculator.
| -4 -3 -1:-12 |
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |
Multiply R 1 by -1/4
| 1 3/4 1/4: 3|
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |
add 5 times the 1st row to the 2nd row
| 1 3/4 1/4: 3|
| 0 39/4 21/4: -9|
| -3 -1 -6:-22 |
add 3 times the 1st row to the 3rd row
| 1 3/4 1/4: 3|
| 0 39/4 21/4: -9|
| 0 5/4 -21/4:-13|
multiply the R 2 by 4/39
| 1 3/4 1/4: 3|
| 0 1 7/13: -12/13|
| 0 5/4 -21/4:-13|
add -5/4 times the 2nd row to the 3rd row
| 1 3/4 1/4: 3|
| 0 1 7/13: -12/13|
| 0 0 -77/13:-154/13|
multiply the R 3 by -13/77
| 1 3/4 1/4: 3|
| 0 1 7/13: -12/13|
| 0 0 1: 2|
add -7/13 times the 3rd row to the 2nd row
| 1 3/4 1/4: 3|
| 0 1 0: -2|
| 0 0 1: 2|
add -1/4 times the 3rd row to the 1st row
| 1 3/4 0: 5/2|
| 0 1 0: -2|
| 0 0 1: 2|
add -3/4 times the 2nd row to the 1st row
| 1 0 0: 4|
| 0 1 0: -2|
| 0 0 1: 2|
This is the final triangular form of the Augmented Matrix (the row echelon form):
| 1 0 0: 4|
| 0 1 0: -2|
| 0 0 1: 2|
The solution to the three simultaneous equations is:
From R 1, x = 4
From R 2, y = -2
From R 3, z = 2
The final answer is:
x = 4
y = -2
z = 2
This answer can be verified by substituting the numerical values of x, y and z into the original equations:
-4x - 3y - z = -12
-4*4 - 3*(-2) - 2 = -12, correct
-5x + 6y + 4z = -24
-5*4 + 6*(-2) + 4*2 = -24, correct
-3x - y - 6z = -22
-3*4 - (-2) - 6*2 = -22, correct
Since the numerical values of x, y, and z work in all three of the original equations, the solutions are correct.
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