ExamplesSection 8.1 – Systems of Linear Equations

{ 2𝑥+𝑦=5−4 𝑥+6 𝑦=−2

{ 𝑥+𝑦+𝑧=03 𝑥−2 𝑦 +4 𝑧=9𝑥− 𝑦−𝑧=0

Solutions are found at the intersection of the equations in the system.

Types of SolutionsSection 8.1 – Systems of Linear Equations

Consistent SystemOne solution

Consistent SystemInfinite solutions

Inconsistent SystemNo solution

Using Matrices to Solve Systems of EquationsSection 8.2 – Systems of Linear Equations - Matrices

2 57 1

Matrix – a rectangular array of numbers−3 914 −615 25

1 4 8−3 5 52 7 11

Identifying the Entries in a MatrixEntries are lower case letters with subscripts .The first subscript refers to the row of the entry.The second subscript refers to the column of the entry.

−3 914 −615 25

𝑎21 This refers to the entry in the 2nd row, 1st column. 𝑎32 𝑎12¿25 ¿9

Examples

Using Matrices to Solve Systems of EquationsSection 8.2 – Systems of Linear Equations - Matrices

Augmented Matrix – a matrix that is used to solve a system of equations.

Augmented matrix

{ 2𝑥+𝑦=5−4 𝑥+6 𝑦=−2 { 𝑥+𝑦+𝑧=0

3 𝑥−2 𝑦 +4 𝑧=9𝑥− 𝑦−𝑧=0

264512

011194230111

Augmented matrix

Using Matrices to Solve Systems of EquationsSection 8.2 – Systems of Linear Equations - Matrices

Given the augmented matrix, write the system of equations.

5 𝑥− 𝑦=9

782915

12731135028263

System of Equations

3 𝑥+6 𝑦−2𝑧=−82 𝑥+0 𝑦+5 𝑧=13𝑥+3 𝑦−7 𝑧=12

System of Equations

2 𝑥+8 𝑦=7

3 𝑥+6 𝑦−2𝑧=−82 𝑥+5 𝑧=13𝑥+3 𝑦−7 𝑧=12

Using Matrices to Solve Systems of EquationsSection 8.2 – Systems of Linear Equations - Matrices

The use of Elementary Row Operations is required when solving a system of equations using matrices.

12731135028263

𝑅1↔𝑅3

Elementary Row OperationsI. Interchange two rows.II. Multiply one row by a nonzero number.III. Add a multiple of one row to a different row.

𝑅3→2𝑟38263

1350212731

1641261350212731

1641261350212731

𝑅2→−2𝑟1+𝑟216412611196012731

Using Matrices to Solve Systems of EquationsSection 8.2 – Systems of Linear Equations - Matrices

The solution to the system of equations is complete when the augmented matrix is in Row Echelon Form.

Row Echelon Form

610035102731

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.

810541

0000117105741

Using Matrices to Solve Systems of EquationsSection 8.2 – Systems of Linear Equations - Matrices

Reduced Row Echelon Form

710030109001

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.

610401

0000120104001

A matrix is in reduced row echelon form (rref) when it satisfies the following conditions.

Use matrices to solve the following systems of equations. (pg. 569 #58)Section 8.2 – Systems of Linear Equations - Matrices

12215154324111

𝑅2→−2𝑟1+𝑟2

𝑦−2𝑧=7𝑧=−2

𝑥− 𝑦+𝑧=−42 𝑥−3 𝑦+4 𝑧=−155 𝑥+𝑦−2 𝑧=12

𝐴𝑢𝑔𝑚𝑒𝑛𝑡𝑒𝑑𝑚𝑎𝑡𝑟𝑖𝑥

𝑅3→−5𝑟1+𝑟 3

1221572104111

3276072104111

𝑅2→−𝑟2

3276072104111

𝑅3→−6𝑟 2+𝑟 3

1050072104111

𝑅3→

15 𝑟3

210072104111

𝑥− 𝑦+𝑧=−4 𝑦−2(−2)=7

𝑥−(3)+(−2)=−4𝑦=3

𝑥=1(1 ,3 ,−2)

𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 :

𝑟𝑜𝑤 h𝑒𝑐 𝑒𝑙𝑜𝑛 𝑓𝑜𝑟𝑚(𝑥 , 𝑦 , 𝑧 )

Use matrices to solve the following systems of equations.Section 8.2 – Systems of Linear Equations - Matrices

𝑅1→𝑟 2+𝑟1

𝑦=3𝑧=−2

𝑥− 𝑦+𝑧=−42 𝑥−3 𝑦+4 𝑧=−155 𝑥+𝑦−2 𝑧=12

𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑒 𝑡𝑜𝑅𝑒𝑑𝑢𝑐𝑒𝑑𝑅𝑜𝑤 h𝐸𝑐 𝑒𝑙𝑜𝑛𝐹𝑜𝑟𝑚

𝑅2→2𝑟 3+𝑟2𝑅1→𝑟 3+𝑟1

h𝐴𝑛𝑜𝑡 𝑒𝑟 𝑜𝑝𝑡𝑖𝑜𝑛 :

210072104111

𝑥=1

(1 ,3 ,−2)

𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 :

𝑟𝑒𝑑𝑢𝑐𝑒𝑑𝑟𝑜𝑤 h𝑒𝑐 𝑒𝑙𝑜𝑛 𝑓𝑜𝑟𝑚

(𝑥 , 𝑦 , 𝑧 )

210072103101

210072101001

210030101001

Consistent or Inconsistent System?Section 8.2 – Systems of Linear Equations - Matrices

210050101001

𝑦+3 𝑧=2𝑧=𝑎𝑛𝑦 𝑟𝑒𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟

𝑂𝑛𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 :𝐶𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚

200070104001

000023102501 𝑥+5𝑧=2

(𝑥=2−5 𝑧 , 𝑦=2−3 𝑧 ,𝑧=𝑎𝑛𝑦𝑟𝑒𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟 )

𝑦=5𝑧=2

𝑥=1

𝑦=70=2

𝑥=4𝑁𝑜𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 : 𝐼𝑛𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚

𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 :𝐶𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚

{(𝑥 , 𝑦 ,𝑧 )∨𝑥=2−5 𝑧 , 𝑦=2−3 𝑧 , 𝑧=𝑎𝑛𝑦 𝑟𝑒𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟 }𝑜𝑟

Use matrices to solve the following systems of equations. (pg. 569 #66)Section 8.2 – Systems of Linear Equations - Matrices

433162123121

𝑅2→−2𝑟1+𝑟2

0=1

𝑥+2 𝑦− 𝑧=32 𝑥− 𝑦+2 𝑧=6𝑥−3 𝑦+3 𝑧=4

𝐴𝑢𝑔𝑚𝑒𝑛𝑡𝑒𝑑𝑚𝑎𝑡𝑟𝑖𝑥

𝑅3→−𝑟1+𝑟 3 𝑅2→−15 𝑟2

𝑅3→5𝑟2+𝑟 3

𝑁𝑜𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 : 𝑖𝑛𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚

433104503121

145004503121

14500103121

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10000103121

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