R.RosnawatiR.Rosnawati

Any straight line in xy-plane can berepresented algebraically by an equationof the form: ax + by = c

General form: define a linear equation inthe n variables :

◦ Where and b are real constants.◦ The variables in a linear equation are sometimes

called unknowns.

Any straight line in xy-plane can berepresented algebraically by an equationof the form: ax + by = c

General form: define a linear equation inthe n variables :

◦ Where and b are real constants.◦ The variables in a linear equation are sometimes

called unknowns.

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The equations andare linear.

Observe that a linear equation does not involve anyproducts or roots of variables. All variables occuronly to the first power and do not appear asarguments for trigonometric, logarithmic, orexponential functions.

The equations sin x +cos x = 1, log x = 2are not linear.

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1,73 zxyyx

732 4321 xxxx The equations and

are linear. Observe that a linear equation does not involve any

products or roots of variables. All variables occuronly to the first power and do not appear asarguments for trigonometric, logarithmic, orexponential functions.

The equations sin x +cos x = 1, log x = 2are not linear.

Find the solution of

Solution(a)we can assign an arbitrary value to x and solve

for y , or choose an arbitrary value for y and solvefor x .If we follow the first approach and assign xan arbitrary value ,we obtain

◦ arbitrary numbers are called parameter.◦ for example

124)a( yx Find the solution of

Solution(a)we can assign an arbitrary value to x and solve

for y , or choose an arbitrary value for y and solvefor x .If we follow the first approach and assign xan arbitrary value ,we obtain

◦ arbitrary numbers are called parameter.◦ for example

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◦ A solution of a linear equation is a sequence of n numberss1, s2, ..., sn such that the equation is satisfied. The set ofall solutions of the equation is called its solution set orgeneral solution of the equation

◦ A solution of a linear equation is a sequence of n numberss1, s2, ..., sn such that the equation is satisfied. The set ofall solutions of the equation is called its solution set orgeneral solution of the equation

Find the solution of

Solution(b)we can assign arbitrary values to any two

variables and solve for the third variable.◦ for example

◦ where s, t are arbitrary values

.574(b) 321 xxx Find the solution of

Solution(b)we can assign arbitrary values to any two

variables and solve for the third variable.◦ for example

◦ where s, t are arbitrary valuestxsxtsx 321 ,,745

A finite set of linear equationsin the variablesis called a system of linear

equations or a linear system .

A sequence of numbersis called a

solution of the system.

A system has no solution issaid to be inconsistent ; ifthere is at least one solutionof the system, it is calledconsistent.

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bxaxaxa

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A finite set of linear equationsin the variablesis called a system of linear

equations or a linear system .

A sequence of numbersis called a

solution of the system.

A system has no solution issaid to be inconsistent ; ifthere is at least one solutionof the system, it is calledconsistent.

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nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

...

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2211

22222121

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An arbitrary system of mlinear equations in n unknowns

Every system of linear equations haseither no solutions, exactly onesolution, or infinitely many solutions.

A general system of two linearequations: (Figure1.1.1)

◦ Two lines may be parallel -> nosolution

◦ Two lines may intersect at only one point-> one solution

◦ Two lines may coincide-> infinitely many solution

Every system of linear equations haseither no solutions, exactly onesolution, or infinitely many solutions.

A general system of two linearequations: (Figure1.1.1)

◦ Two lines may be parallel -> nosolution

◦ Two lines may intersect at only one point-> one solution

◦ Two lines may coincide-> infinitely many solution

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The location of the +’s,the x’s, and the =‘s canbe abbreviated by writingonly the rectangular arrayof numbers.

This is called theaugmented matrix for thesystem.

Note: must be written inthe same order in eachequation as the unknownsand the constants must beon the right.

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The location of the +’s,the x’s, and the =‘s canbe abbreviated by writingonly the rectangular arrayof numbers.

This is called theaugmented matrix for thesystem.

Note: must be written inthe same order in eachequation as the unknownsand the constants must beon the right.

1th column

1th row

The basic method for solving a system of linear equationsis to replace the given system by a new system that hasthe same solution set but which is easier to solve.

Since the rows of an augmented matrix correspond to theequations in the associated system. new systems isgenerally obtained in a series of steps by applying thefollowing three types of operations to eliminate unknownssystematically. These are called elementary rowoperations.1. Multiply an equation through by an nonzero constant.2. Interchange two equation.3. Add a multiple of one equation to another.

The basic method for solving a system of linear equationsis to replace the given system by a new system that hasthe same solution set but which is easier to solve.

Since the rows of an augmented matrix correspond to theequations in the associated system. new systems isgenerally obtained in a series of steps by applying thefollowing three types of operations to eliminate unknownssystematically. These are called elementary rowoperations.1. Multiply an equation through by an nonzero constant.2. Interchange two equation.3. Add a multiple of one equation to another.

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The solution x=1,y=2,z=3 is now evident.