At least two linear equations in more than two variables.
5x + 3y - 8z = 11
-x + 7y = 10
9x + z = 0
Coefficient Matrix
A matrix consisting of the coefficients of the terms from a system of equations.
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| Coefficient Matrix |
Augmented Matrix
A matrix consisting of the coefficients and constants of the terms from a system of equations.
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| Augmented Matrix |
Side by Side comparison
Gaussian Elimination
Gaussian elimination is the process of using elementary row operations to change an augmented matrix to row-echelon form or reduced row-echelon form.
You can perform the following row operations to any matrix
- Swap any rows
- Multiply any row by a nonzero scalar
- Replace a row with the sum of two rows
Row-Echelon Form
A matrix is in row echelon form if- Rows consisting entirely of zeros (if any) appear at the bottom of the matrix
- The first entry in a row with nonzero entries is 1, called a leading 1.
- For two consecutive rows with nonzero entries, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.
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| Row-echelon form |
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| Row-echelon form |
Notice in each example, the triangular nature of the nonzero entries.
Reduced Row-Echelon Form
A matrix is in row reduced echelon form if the following conditions are satisfied:
- The first nonzero element in each row (if any) is a "1" (a leading coefficient).
- Each leading coefficient is the only nonzero element in its column.
- All the all-zero rows (if any) are at the bottom of the matrix.
- The leading coefficients form a "stairstep pattern" from northwest to southeast.
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| reduced row-echelon |
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| reduced row-echelon |
Solving a System Using Gaussian Elimination
Example: Write the system of equations in an augmented matrix. Use elementary row operations to get the matrix into row-echelon form or reduced-row echelon form to solve the system.3x + 5y = 7
6x - y = -8
Practice: Write the system of equations in an augmented matrix. Use elementary row operations to get the matrix into row-echelon form or reduced-row echelon form to solve the system.
#1) 5x - 2y = 5
x + y = 8
#2) 2x + y - 2z - 7 = 0
x - 2y - 5z + 1 = 0
4x + y + z + 1 = 0
Homework
Solving a System Using Gaussian Elimination
Write the system of equations in an augmented matrix. Use elementary row operations to get the matrix into row-echelon form or reduced-row echelon form to solve the system.#1) 3x + 5y = 7
6x - y = -8
#2) 4x - 7y = -2
x + 2y = 7
#3) 3x + 3y = -9
-2x + y = -4
#4) 5x = 3y - 50
2y = 1 - 3x
#5) x - y + z = 3
2y - z = 1
-x + 2y = -1
#6) x + y + z = -2
2x + 3y + z = -11
-x + 2y - z = 8
#7) 2x + 6y + 8z = 5
-2x + 9y - 12z = -1
4x + 6y - 4z = 3
#8) x + y + z - 6 = 0
2x - 3y + 4z - 3 = 0
4x - 8y + 4z - 12 = 0
#9) x + 2y = 5
3x + 4z = 2
3w + 2y = -2
-2w + 3z = 1
Answers Solving a System Using Gaussian Elimination
#1) (-1, 2)#2) (3, 2)
#3) (1/3, -10/3)
#4) (-97/19, 155/19)
#5) (3, 1, 1)
#6) (-13, 2, 9)
#7) (1/2, 1/3, 1/4)
#8) (7, 1, -2)
#9) (-61/35, 62/35, 113/70, -29/35)
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