1. Section 8.1 – Systems of Linear Equations
Examples
2𝑥+𝑦=5 −4𝑥+6𝑦=−2
𝑥+𝑦+𝑧=0 3𝑥−2𝑦+4𝑧=9 𝑥−𝑦−𝑧=0
Solutions are found at the intersection of the equations in the system.

2. Section 8.1 – Systems of Linear Equations
Types of Solutions
Consistent System
Consistent System
Inconsistent System
One solution
Infinite solutions
No solution

3. Section 8.2 – Systems of Linear Equations - Matrices
Using Matrices to Solve Systems of Equations
Matrix – a rectangular array of numbers
− −
1 4 8 −
Identifying the Entries in a Matrix
Entries are lower case letters with subscripts ( 𝑎 𝑚𝑛 ).
The first subscript refers to the row of the entry.
The second subscript refers to the column of the entry.
Examples
− −
𝑎 21
This refers to the entry in the 2nd row, 1st column.
𝑎 21 =14
𝑎 32
=25
𝑎 12 =9

4. Section 8.2 – Systems of Linear Equations - Matrices
Using Matrices to Solve Systems of Equations
Augmented Matrix – a matrix that is used to solve a system of equations.
𝑥+𝑦+𝑧=0 3𝑥−2𝑦+4𝑧=9 𝑥−𝑦−𝑧=0
2𝑥+𝑦=5 −4𝑥+6𝑦=−2
Augmented matrix
Augmented matrix

5. Section 8.2 – Systems of Linear Equations - Matrices
Using Matrices to Solve Systems of Equations
Given the augmented matrix, write the system of equations.
System of Equations
System of Equations
5𝑥−𝑦=9
3𝑥+6𝑦−2𝑧=−8
2𝑥+8𝑦=7
2𝑥+0𝑦+5𝑧=13
𝑥+3𝑦−7𝑧=12
3𝑥+6𝑦−2𝑧=−8
2𝑥+5𝑧=13
𝑥+3𝑦−7𝑧=12

6. Section 8.2 – Systems of Linear Equations - Matrices
Using Matrices to Solve Systems of Equations
The use of Elementary Row Operations is required when solving a system of equations using matrices.
Elementary Row Operations
I. Interchange two rows.
II. Multiply one row by a nonzero number.
III. Add a multiple of one row to a different row.
𝑅 1 ↔ 𝑅 3
𝑅 3 → 2𝑟 3
𝑅 2 → −2𝑟 1 + 𝑟 2

7. Section 8.2 – Systems of Linear Equations - Matrices
Using Matrices to Solve Systems of Equations
The solution to the system of equations is complete when the augmented matrix is in Row Echelon Form.
Row Echelon Form
A matrix is in row echelon form (ref) when it satisfies the following conditions.
The first non-zero element in each row, called the leading entry, is 1.
Each leading entry is in a column to the right of the leading entry in the previous row.
Rows with all zero elements, if any, are below rows having a non-zero element.

8. Section 8.2 – Systems of Linear Equations - Matrices
Using Matrices to Solve Systems of Equations
Reduced Row Echelon Form
A matrix is in reduced row echelon form (rref) when it satisfies the following conditions.
The matrix is in row echelon form (i.e., it satisfies the three conditions listed for row echelon form.
The leading entry in each row is the only non-zero entry in its column.

9. Section 8.2 – Systems of Linear Equations - Matrices
Use matrices to solve the following systems of equations. (pg. 569 #58)
𝐴𝑢𝑔𝑚𝑒𝑛𝑡𝑒𝑑 𝑚𝑎𝑡𝑟𝑖𝑥
𝑥−𝑦+𝑧=−4
2𝑥−3𝑦+4𝑧=−15
𝑅 2 → −2𝑟 1 + 𝑟 2
5𝑥+𝑦−2𝑧=12
𝑅 3 → −5𝑟 1 + 𝑟 3
𝑅 2 → −𝑟 2
𝑅 3 → −6𝑟 2 + 𝑟 3
𝑅 3 → 𝑟 3
𝑥−𝑦+𝑧=−4
𝑦−2(−2)=7
𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛:
𝑦−2𝑧=7
𝑦=3
(𝑥, 𝑦,𝑧)
𝑧=−2
𝑥−(3)+(−2)=−4
(1, 3,−2)
𝑟𝑜𝑤 𝑒𝑐ℎ𝑒𝑙𝑜𝑛 𝑓𝑜𝑟𝑚
𝑥=1

10. Section 8.2 – Systems of Linear Equations - Matrices
Use matrices to solve the following systems of equations.
𝐴𝑛𝑜𝑡ℎ𝑒𝑟 𝑜𝑝𝑡𝑖𝑜𝑛:
𝑥−𝑦+𝑧=−4
𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑒 𝑡𝑜 𝑅𝑒𝑑𝑢𝑐𝑒𝑑 𝑅𝑜𝑤 𝐸𝑐ℎ𝑒𝑙𝑜𝑛 𝐹𝑜𝑟𝑚
2𝑥−3𝑦+4𝑧=−15
5𝑥+𝑦−2𝑧=12
𝑅 1 → 𝑟 2 + 𝑟 1
𝑅 1 → 𝑟 3 + 𝑟 1
𝑅 2 → 2𝑟 3 + 𝑟 2
𝑥=1
𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛:
𝑦=3
(𝑥, 𝑦,𝑧)
𝑧=−2
(1, 3,−2)
𝑟𝑒𝑑𝑢𝑐𝑒𝑑 𝑟𝑜𝑤 𝑒𝑐ℎ𝑒𝑙𝑜𝑛 𝑓𝑜𝑟𝑚

11. Section 8.2 – Systems of Linear Equations - Matrices
Consistent or Inconsistent System?
𝑥=1
𝑦=5
𝑂𝑛𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛:𝐶𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚
𝑧=2
𝑥=4
𝑦=7
𝑁𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛:𝐼𝑛𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚
0=2
𝑥+5𝑧=2
𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠:𝐶𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚
𝑦+3𝑧=2
𝑧=𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟
(𝑥=2−5𝑧,𝑦=2−3𝑧,𝑧=𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟) 𝑜𝑟
{(𝑥, 𝑦,𝑧)|𝑥=2−5𝑧,𝑦=2−3𝑧,𝑧=𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟}

12. Section 8.2 – Systems of Linear Equations - Matrices
Use matrices to solve the following systems of equations. (pg. 569 #66)
𝐴𝑢𝑔𝑚𝑒𝑛𝑡𝑒𝑑 𝑚𝑎𝑡𝑟𝑖𝑥
𝑥+2𝑦−𝑧=3
2𝑥−𝑦+2𝑧=6
𝑅 2 → −2𝑟 1 + 𝑟 2
𝑥−3𝑦+3𝑧=4
𝑅 3 → −𝑟 1 + 𝑟 3
𝑅 2 → − 𝑟 2
𝑅 3 → 5𝑟 2 + 𝑟 3
0=1
𝑁𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛:𝑖𝑛𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚