Linear Equation Systems: Building, Solving, and Understanding, Slides of Engineering Mathematics

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Linear Equation System

Outline

• Brief Review & Example
• Homework/Exam's Solution
• Q&A (10min left)

Q1: How to build up a linear equation system?

Example 1: Nutrition Problem

Find a combination of food A, B, C and D in order to satisfy the nutrition requirement exactly. Let xA, xB, xC and xD be the amount of food A, B, C and D respectively.

Food A Food B Food C Food D Requirement

Protein 9 8 3 3 5

Carbohydrate 15 11 1 4 5

Vitamin A 0.02 0.003 0.01 0.006 0.

Vitamin C 0.01 0.01 0.005 0.05 0.

Q1: How to build up a linear equation system?

Example 1: Nutrition Problem

Find a combination of food A, B, C and D in order to satisfy the nutrition requirement exactly. Let xA, xB, xC and xD be the amount of food A, B, C and D respectively.

Food A Food B Food C Food D Requirement

Protein 9 8 3 3 5

Carbohydrate 15 11 1 4 5

Vitamin A 0.02 0.003 0.01 0.006 0.

Vitamin C 0.01 0.01 0.005 0.05 0.

A

B

C

D

x

x

x

x

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Q1: How to build up a linear equation system?

Example 2: Electrical Network Problem

Find a combination of currents i1, i2, and i in order to satisfy the Kirchhoff Law exactly.

Kirchhoff’s Current Law (KCL)

At any point of a circuit, the sum of the inflowing

currents equals the sum of the outflowing currents.

Kirchhoff’s Voltage Law (KVL)

In any closed loop, the sum of all voltage drops equals the

impressed electromotive force.

1st equation: Node Q i i 1 - 2 (^) + =0 i 3

2st equation: Node P - + i 1 (^) i 2 (^) - =0 i 3

3st equation: Right loop

4st equation: Left loop

10 i 2 (^) +25 =90 i 3

20 -10 i 1 (^) i 2 =

Q1: How to build up a linear equation system?

Example 2: Electrical Network Problem

Find a combination of currents i1, i2, and i in order to satisfy the Kirchhoff Law exactly.

Kirchhoff’s Current Law (KCL)

At any point of a circuit, the sum of the inflowing

currents equals the sum of the outflowing currents.

Kirchhoff’s Voltage Law (KVL)

In any closed loop, the sum of all voltage drops equals the

impressed electromotive force.

1 2 3

1 2 3

2 3

1 2

• • =
• • • =

10 +25 =

20 +10 =

i i i

i i i

i i

i i

1

2

3

1 -1 1 0

-1 1 -1 0

0 10 25 90

20 10 0 80

i

i

i

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Q1: How to build up a linear equation system?

Example 3: Balancing a chemical equation

please find a combination of x1, x2, x3 and x4, such that the numbers of atoms of carbon (C), hydrogen (H), and oxygen (O) are the same on both sides of this reaction, in which propane and give carbon dioxide and water.

x C H 1 3 8 (^) + x O 2 2 (^)  x CO 3 2 (^) + x H O 4 2

1st equation: balancing carbon

2st equation: balancing hydrogen

3st equation: balancing oxygen

3 x 1 = x 3

8 x 1 =2 x 4

x 2 = x 4

Q1: How to build up a linear equation system?

Example 3: Balancing a chemical equation

please find a combination of x1, x2, x3 and x4, such that the numbers of atoms of carbon (C), hydrogen (H), and oxygen (O) are the same on both sides of this reaction, in which propane and give carbon dioxide and water.

x C H 1 3 8 (^) + x O 2 2 (^)  x CO 3 2 (^) + x H O 4 2

3 x 1 = x 3

8 x 1 =2 x 4

x 2 = x 4

1 1

2 2

3 3

4 4

x x

x x

x x

x x

11 12 1 1 1 11 12 1 1

21 22 2 2 2 21 22 2 2

1 2 1 2

=

n n

n n

m m mn n n m m mn n

a a a x b a a a b

a a a x b a a a b or

a a a x b a a a b

       

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       

       

               

Q2: What will the solution set

of a linear equation system look like?

solution set: all possible solution of x

a) infinitely many solutions

b) single unique solution

c) no solution

Q2: What will the solution set

of a linear equation system look like?

Geometric Thinking:

1. No solution if the lines are parallel

2. Precisely one solution if the lines intersect

3. Infinitely many solutions if the lines coincide

 ( , h m^ ) :^^ h^ ^ 8,^ m ^4 

 ( , h m^ ) :^^ h^ ^8 

( , h m^ ) :^^ h^ ^ 8,^ m ^4 

1 -1 1 0

-1 1 -1 0

0 10 25 90

20 10 0 80

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Q3: How to solve a linear equation system?

Example 1:

Example 2:

(^) Example 3:

• Step 0: Write the linear system in matrix

format

• Step 1: Try to transform the matrix into

upper triangular form

• Step 2: Solve for the variables one by one,

in backward order

Q3: How to solve a linear equation system?

Example 2: Electrical Network Problem

1 -1 1 0

-1 1 -1 0

0 10 25 90

20 10 0 80

           

Q3: How to solve a linear equation system?

1 -1 1 0

0 0 0 0

0 10 25 90

0 30 -20 80

           

1st step: elimination of x

Use the first equation to eliminate x1 in other equations

1. Add 1 times the first equation to the second equation

2. Add -20 times the first equation to the fourth equation

Example 2: Electrical Network Problem

Q3: How to solve a linear equation system?

1 -1 1 0

0 0 0 0

0 10 25 90

0 30 -20 80

 

 

 

 

   

2st step: elimination of x

1. Put 0=0 at the end and move the third equation and the fourth equation one place up

2. Add -3 times the second equation to the third equation

1 -1 1 0

0 10 25 90

0 30 -20 80

0 0 0 0

           

1 -1 1 0

0 10 25 90

0 0 -95 -

0 0 0 0

           

upper triangular form