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Three variables
Systems of Equations and Inequalities
Matrices and Systems of Equations
We can translate a given system of equations into an augmented matrix.
With two rows and two columns, this matrix is a 2x3 matrix.
2 3 7
4 2
x y
x y
Gaussian Elimination
To solve a system of equations using Gaussian elimination with matrices, we use the same rules as before.
1. Interchange any two rows.2. Multiply each entry in a row by the same
nonzero constant.3. Add a nonzero multiple of one row to
another row.
Example 1
Solve:
Matrix:
2 4 3
2 10 6
3 4 7
x y x
x y z
x z
2 1 4 3
1 2 10 6
3 0 4 7
1 2R R
1 2 10 6
2 1 4 3
3 0 4 7
1 2 22*R R R
1 2 10 6
0 3 24 9
3 0 4 7
1 3 33R R R
1 2 10 6
0 3 24 9
0 6 34 25
2 23R R
1 2 10 6
0 1 8 3
0 6 34 25
2 1 12R R R 1 0 6 0
0 1 8 3
0 6 34 25
2 3 36R R R
1 0 6 0
0 1 8 3
0 0 14 7
3 314R R
1 2 10 6
0 1 8 3
0 0 1 0.5
Rewrite system
and back substitute.
3,7, 0.5
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
1 1 2 0
1 2 3 1
2 2 1 3
1 2 2
1 1 2
0 1 1
0
1
1
2 3 1
R R R
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
1 2 2
1 1 2
0 1 1
0
1
1
2 3 1
R R R
0 1 1
1 1 2 0
2 3
1
2 1
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
1 1 2 0
0 1 1 1
2 2 1 3
1 3 3
2 2 4 0
2 2 1 3
3
2
0 0 3
R R R
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
1 3 3
2 2 4 0
2 2 1 3
3
2
0 0 3
R R R
0 0
1 1 2 0
0 1 1
3
1
3
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
1 1 2 0
0 1 1 1
0 0 3 3
2 2
0
1
1 1 1
R R
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
2 2
0
1
1 1 1
R R
0 1 1 1
1 1 2 0
0 0 3 3
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
1 1 2 0
0 1 1 1
0 0 3 3
2 1 1
0 1 1 1
1
1 0 1 1
1 2 0
R R R
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
2 1 1
0 1 1 1
1
1 0 1 1
1 2 0
R R R
0 1 1 1
0 0 3 3
1 0 1 1
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
1 0 1 1
0 1 1 1
0 0 3 3
3 3
1
30 0 1 1
R R
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
3 3
1
30 0 1 1
R R 1 0 1 1
0 1 11
0 0 1 1
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
1 0 1 1
0 1 11
0 0 1 1
3 2 2
0 0 1 1
0 1 1
1
1
0 0 2
R R R
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
3 2 2
0 0 1 1
0 1 1
1
1
0 0 2
R R R
1 0 1 1
0 0 1 1
0 1 0 2
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
1 0 1 1
0 1 0 2
0 0 1 1
3 1 1
0
1 0
0 1
0 0
1
1 0 1 1
R R R
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
3 1 1
0
1 0
0 1
0 0
1
1 0 1 1
R R R
0 1 0 2
0 0 1 1
1 0 0 0
Example
• Solve the system of equations using Gauss-Jordan Method
2 0
2 3 1
2 2 3
x y z
x y z
x y z
1 0 0 0
0 1 0 2
0 0 1 1
(0, 2, 1)
Systems of Equations: MatricesDefinition: An m X n matrix is a rectangular array of
numbers with m rows and n columns.
The numbers
are the entries of the matrix.
The subscript on the entry indicates that it is in the ith row and the jth column
mnmm
n
n
aaa
aaa
aaa
21
22221
11211
ija
ija
Augmented Matrix
114
03
523
zx
zyx
zyx
11401
0131
5123
Linear System Augmented Matrix
Elementary Row Operations
1. Add a multiple of one row to another.
2. Multiply a row by a nonzero constant.
3. Interchange two rows.
ijiRkRR
ikR
jiRR
Symbol Description
Change the ith row by adding k times row j to row i, putting the result back in row i.
Multiply the ith row by k.
Interchange row i and row j.
Example
Solve:
Matrix:
2 4 3
2 10 6
3 4 7
x y x
x y z
x z
2 1 4 3
1 2 10 6
3 0 4 7
1 2R R
1 2 10 6
2 1 4 3
3 0 4 7
1 2 22*R R R
1 2 10 6
0 3 24 9
3 0 4 7
1 3 33R R R
1 2 10 6
0 3 24 9
0 6 34 25
2 23R R
2 1 12R R R
1 0 6 0
0 1 8 3
0 6 34 25
2 3 3
6R R R
1 0 6 0
0 1 8 3
0 0 14 7
3 3
14R R
1 2 10 6
0 1 8 3
0 0 1 0.5
Rewrite system
and back substitute. 3,7, 0.5
50100
3810
0601
.
Row-Echelon Form and Reduced Row-Echelon Form
A matrix is in row-echelon form if it satisfies the following conditions.
1. The first nonzero entry in each row (left to right) is 1. This is called a leading 1.
2. The leading entry in each row is to the right of the leading entry in the row immediately above it.
3. Every number above and below each leading entry is a zero. This is called reduced row-echelon form.
Inconsistent and Dependent Systems
A leading variable is a linear system is one that corresponds to a leading entry in the row-echelon form of the matrix of the system.
Suppose the system has been transformed into row-echelon form. Then exactly one of the following is true.
1. No solution. There is a row that represents 0 = C, where C is not zero. The system has no solution and is inconsistent.
2. One solution. If each variable is a leading variable, then the system has exactly one solution.
3. Infinitely many solutions. If there is at least one row of all zeros, the system has infinitely many solutions. The system is called dependent.
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