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Lecture #1 – Tuesday, January 6, 2004

1.1 SOLUTIONS AND ELEMENTARY OPERATIONS

Example

Create a diet from fish and meal that contains 193g of proteins and 83g of carbohydrate. We know that fish

contains 70% protein and 10% carbohydrate, and meal contains 30% protein and 60% carbohydrate.

• Assume that the diet contains xg of fish and yg of meal, we obtain

=+

=+

836.01.0

1933.07.0

yx

yx .

Definition

A linear equation is an equation of the form bxaxann =++ ...

11 where:

• x1,…, xn are variables;

• a1,…, an are real numbers called coefficients;

• b is the constant term.

Examples

1) cbyax=+ .

2) 023 321 =−+ xxx .

3) 12 2

2

1=+ xx – not a linear equation.

Definition

A finite collection of linear equations in the variables x1,…, xn is called a system of linear equations.

Examples

1)

=

=

3

1

2

1

x

x.

2)

=−

=+

03

1442

21

21

xx

xx

Definition

Given a linear equation bxaxann =++ ...

11 , a sequence of n real numbers s1,…, sn is called a solution to the

linear equation if bsasann =++ ...

11 . Similarly, this also applies to a given system of linear equations.

Example

Given

( )

=

=

3

1

2

1

1x

x

S and

( )

=−

=+

03

1442

21

21

2xx

xx

S, what is the solution to (S1) and (S2)?

• (1, 3) is a solution of (S1).

• (1, 3) is also a solution of (S2) because

(

)

(

)

143412 =+ and

(

)

(

)

0313 =− .

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Example

•

( )

=+

=+

2

1

3yx

yx

S has no solution.

Example

Prove that for any s, t in R,

−=

=

=

−+=

ss

ts

ss

tss

2

2

1

2

3

4

3

2

1

is a solution to the system

=+++

=−+−

2

13

4321

4321

xxxx

xxxx .

• 13

2

1

2

3

2

2

1

3

2

3

34321 =+−+−+−=

−−+−−+=−+− ttsssststsssss .

• 22

2

1

2

3

2

2

1

2

3

4321 =+−−+++=−+++−+=+++ ttsssststsssss .

Definition

s, t are called parameters. s1,…, s4 described this way is said to be given in parametric form and is called the

general solution of the system.

Remarks

• When only 2 variables are involved, solution to

systems of linear equations can be described

geometrically because a linear equation

cbyaxL=+: is a straight line if a, b are not

both 0.

(

)

21 ,ssP is in L if it is a solution of

cbyax=+ .

• If there are two linear equations, cbyaxL=+:

1 and feydxL=+:

2, then the solution to the system

( )

=+

=+

feydx

cbyax

S is the intersection of L1 and L2.

• The solution of (S) is

(

)

21 ,ss where

(

)

2121 ,LLssP∩=.

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• The solution of (S) are given by the

(

)

21 ,ss

such that cbsas=+ 21 (and this implies that

fesds=+ 21 ).

• (S) has no solution.

Definition

The elementary operations are:

1) Interchange 2 equations.

2) Multiply one equation by a non-zero number.

3) Add a multiple of one equation to a different equation.

Definition

Two systems of linear equations are said to be equivalent iff the solutions of the systems are the same.

Theorem

Suppose that an elementary operation is performed on a system of linear equations, then the resulting system

is equivalent to the original one.

Example

Solve:

=−

=+

43

12

yx

yx .

•

−=

=

⇔

−=−=

=

⇔

=

′′′

=+

′′

=

′

=

⇔

′

+=

′′

=

=+

⇔

=

′

=−

=+

⇔

=−

=+

7

1

7

9

7

2

7

9

12

7

9

12

97

97

12

2826

12

43

12

12

21

212

1

22

1

2

1

y

x

y

x

RRyx

RRx

RRRx

Ryx

RRyx

Ryx

Ryx

Ryx

• The solution of the system is

−7

1

,

7

9.

L

1

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