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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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bxaxaxa nn ...2211
,,...,, 21 naaa
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.
,132
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.
nxxx ,...,, 21
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
...
...
...
2211
22222121
11212111
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.
nsss ,...,, 21mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
...
...
...
2211
22222121
11212111
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
zero)bothnot,(
zero)bothnot,(
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11111
bacybxa
bacybxa
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
...
...
...
2211
22222121
11212111
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.
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
...
...
...
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mmnmm
n
n
baaa
baaa
baaa
...
...
...
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222221
111211
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.
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