11 linear equations

8/9/2019 11 Linear Equations

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Tuesday, January 27, 2015

AIRIL AHMAD

LINEAR EQUATIONS

AIRIL AHMAD


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AIRIL AHMAD

Deter!ne t"e #a$ue %& a #ar!a'$e(!#en t"e %t"er #ar!a'$es

Example 1:

Given 6m + n = 27, find the vale !f n, "hen m=#

$!lti!n: 6 m + n = 27

6 + n = 27

%#&

1' + n = 27

n = 27

( 1'

n = )

*


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AIRIL AHMAD

Example 2:

6

3=m

Given 6m + n = 7, find the vale !f m, "hen n=

$!lti!n: 6 m + n = 7

6 m + = 7

%&

6 m + = 7

6m = 7

(

6m = # *

m= #

6

1

2

2

1=m


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AIRIL AHMAD

E)er*!seind the vale !f - ind the vale !f x

1. 2x + #- = 16 , x = 2

2. x + 2- =7 , x = /

#. (#x + 2- = ) , x = (#

. (2x +#- = (# , x = (1

/. 'x + 2- (10 =0 , x =

6. #x + 2- = 1/ , - = #

7. #x +- = ' , - = 2

'. /x + 2- = , - = (#

). 2x ( #- = (# , - = (1

10. 2x + #- (10 =0 , - =


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AIRIL AHMAD

ind the vale !f x

6. #x + 2- = 1/ , - = #

7. #x +- = ' , - = 2

'. /x + 2- = , - = (#

). 2x ( #- = (# , - = (1

10. 2x + #- (10 =0 , - =


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AIRIL AHMAD

E)er*!se

ind the vale !f x

1. #x + 2- = 1/ , - = #1. #x +- = ' , - = 2


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AIRIL AHMAD

E)er*!seind the vale !f -

;34

3+= yx11. x =

12. x = /

1#. x = (#

;6

3

1

5

2=+ yx

;64

3yx =


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AIRIL AHMAD

E)er*!seind the vale !f -

;2432 yxyx +=1. x =

1/. x = (10;4

54 yx =

T d J 27 2015


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AIRIL AHMAD

E)er*!seind the vale !f x

;13

2= xy16. - = #

17. - = 6;3

210

5

3yx =

T d J 27 2015


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AIRIL AHMAD

E)er*!seind the vale !f x

;3824 yx =+1'. - = (2

1). - = (#

20. - = ('

;01028 =+ yx

;124

3

3

2= yx

T d J 27 2015


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AIRIL AHMAD

SIMULTANEOUSLINEAR EQUATIONS

AIRIL AHMAD

T d J 27 2015


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AIRIL AHMAD

Example 1:

a + = 10 and a = 2

a + = 10

a = 2

3%1&

3%2&%+&

2a = 12

a = 12

2

a= 6

4hen a=6 , then:

a + = 10 3.%1& + = 10

6

= 10 ( 6

= *

T d J 27 2015


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AIRIL AHMAD

Example 2:

#m ( n = 7

m + n = /

3%1&

3%2&%+&

m = 12

m = 12

m= #

4hen m=# , then:

m + n = / 3.%2& + n = /

#

n = / ( #

n = 2 *

#m ( n = 7 and m +n =

/

Tuesday January 27 2015


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AIRIL AHMAD

Example #:

2e +#f = 1

/e ( #f = 1#

3%1&

3%2&%+&

7e = 1

e = 1

7

e= 2

4hen e=2 , then:

2e + #f = 1 3.%1&2 + #f = 1

%2&

#f = 1 (

f = (#

*

2e + #f=1 and /e#f = 1#

#

f = (1



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AIRIL AHMAD

Example :

2x + - =

2x - = 16

3%1&

3%2&%+&

x = 20

x = 20

x= /

4hen x=/ , then:

2x + - = 3.%1&2 + - =

%/&

- = ( 10

- = (6*

2x + - = and 2x ( - = 16



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AIRIL AHMAD

E)er*!se

1. x + - = and x - = 22. x + - = ' and x - =

#. #x + - = 20 and x - =

. /x - =1) and 7x + -=17



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AIRIL AHMAD


+SUTRA-TION.

AIRIL AHMAD



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AIRIL AHMAD

Example 1:

2a + =10 and a + =2

2a + = 10

a + = 2

3%1&

3%2&%(&

a = '

4hen a=' , then:

a + = 2 3.%2& + = 2

'

= 2 ( '

= (6*



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AIRIL AHMAD

Example 2:

#m + n = 7

m + n = /

3%1&

3%2&%(&

2m = 2

m = 2

2

m = 1

4hen m=1 , then:

m + n = / 3.%2& + n = /

1

n = / ( 1

n = *

#m + n = 7 and m +n = /



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AIRIL AHMAD

E)er*!se1. /e + #f = 1# and 2e + #f = 12. x + - = 16 and 2x + - =

#. 2x + - = 10 and x + - = 7. 2x + - = 12 and x + - = 7



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AIRIL AHMAD

1. /e + #f = 1# and 2e + #f = 1



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AIRIL AHMAD

SIMULTANEOUS LINEAREQUATIONS

+MULTI/LI-ATION /ART 1.

AIRIL AHMAD



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AIRIL AHMAD

Example 1:

a + = / and 2a + # = 1/

a + = /

2a + # = 1/

3%1&

3%2&

%(&

a = 0

4hen a=0 , then:

a + = / 3.%1& + = /

0

= / ( 0

= /*

)

#a + # = 1/ ....%#&

2a + # = 1/ 3%2&

+.+.

) ) )



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AIRIL AHMAD

Example 2:

a + = # and /a ( ' = 1

a + = #

/a ( ' = 1

3%1&

3%2&

%+&

7a = 7

4hen a=1 , then:

a + = # 3.%1& + = #

1

= # ( 1

= 2

*

) 2

2a + ' = 6 ....%#&

/a ( ' = 1 3%2&

+.+.

) 2 ) 2 ) 2

a = 1

=2

1

4

2=



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AIRIL AHMAD

%#&Example #:

2a + / = 16 and a ( = 1

2a + / = 16

a ( = 1

3%1&

3%2&

%+&

7a = 21

4hen a=# , then:

2a + / = 16 3.%1&

2 + / = 16

6 + / = 16

/ = 10

*

) 5

2a + / = 16 ....%1&

/a ( / = / 3%#&

+.+. ) 5 ) 5 ) 5

a = #

= 25

10=

/ = 16 ( 6



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AIRIL AHMAD

E)er*!se1. 2e + f = (# and #e + f = (72. x + #- = # and x + - = 0

#. x + - = # and #x ( /- = 17. #m ( n = # and m + #n = 11



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AIRIL AHMAD

1. x + - = # and #x ( /- = 17



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uesday, Ja ua y , 0 5

AIRIL AHMAD


+MULTI/LI-ATION /ART2.

AIRIL AHMAD



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y, y ,

AIRIL AHMAD

Example 1:

#a + 2 = ) and 2a + # = 11

#a + 2 = )

2a + # = 11

3%1&

3%2&

%(&

/a = /

4hen a=1 , then:

#a + 2 = ) 3.%1 + 2 = )

%1&

2 = ) ( #

= #*

)

3a 4' 27 6666+.

a 4' 22 8+.

+.+.

) ) )

) 2 ) 2 ) 2) 2

a = 1

# + 2 = )

2 = 6



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y, y ,

AIRIL AHMAD



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y, y ,

AIRIL AHMAD

Example 2:

#a + / = 21 and /a ( # = 1

#a + / = 21

/a ( # = 1

3%1&

3%2&

%+&

#a = 6'

4hen a=2 , then:

#a + / = 21 3.%1 + / = 21

%2&

/ = 21 ( 6

= #*

)

3a 15' 4 6666+.

25a 15' 5 8+.

+.+.

) ) )

) 5 ) 5 ) 5) 5

a = 2

6 + / = 21

/ = 1/



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y, y ,

AIRIL AHMAD

Example #:

2a ( # = and 7a ( = 1

2a ( # =

7a ( = 1

3%1&

3%2&

%(&

(1#a = 1#

4hen a = (1 , then:

2a ( # = 3.%1&

2 ( # =

%(1&

(# = + 2

=

*

)

9a 12' 14 666+.

21a 12' 8+.

+.+.

) ) )

) ) ) )

a = (1

(2 ( # =

(# = 6

3

6

= (2



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y, y ,

AIRIL AHMAD

E)er*!se1. 2e + #f = 7 and /e ( f = 62. 2x + #- = ' and #x ( 2- = (1

#. #x ( 2- = 1 and /x ( #- = #. 2m ( /n = ' and #m ( 7n = 11



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y y

AIRIL AHMAD

SIMULTANEOUS

LINEAR EQUATIONS+/ROLEM SOL:IN;.

AIRIL AHMAD



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y y

AIRIL AHMAD

E)a


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AIRIL AHMAD

E)a


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AIRIL AHMAD

E)a


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AIRIL AHMAD

E)a


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AIRIL AHMAD

E)a


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AIRIL AHMAD

E)a


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AIRIL AHMAD

E)a


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Tuesday, January 27, 2015 AIRIL AHMAD


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4

+2.Example 1:

a + # = # and = 2

a + # = #

= 2

3%1&

3%2&

$titte =2 in 3%1&

a + # = #

a + # = #

a + 6 = #

a = #

a = (#



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+4

+.Example 2:

2a ( # = 6 and a = (#

2a ( # = 6a = (#

3%1&

3%2&

$titte a = (# in 3%1&

2a ( # = 6

2 # = 6

( 6 # = 6

(# = 6

( = 12

= (12



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+2'.Example #:

2a + = 1/ and a = 2

2a + = 1/a = 2

3%1&

3%2&

$titte a = 2 in 3%1&

2a + = 1/

2 + = 1/

+ = 1/

/ = 1/

= 1/

= #

5a 2a 4

+.


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+

+2 '.Example /:

3%1&

3%2&

$titte a = 2 + in 3%1&

a + = 10

+ = 10

2 + + = 10

2 + 2 = 10

2 = 10 ( 2

= '

2

a + = 10 and a = 2

a + = 10

a = 2

a = 2

9- in 2nde;ati!n 3

a = 2

2 = '

= a 2

a 4



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( m

+5 .Example 6:

3%1&

3%2&

$titte n = / ( m in 3%1&

#m ( n = 7

#m ( = 7

#m ( / + m = 7

m / = 7

m = 7 + /

m = 12

4

m + n = /

9- in 2nde;ati!n 3

n = /

m = 12

m = #

#m ( n = 7 and m +n =

/

#m ( n = 7

m + n = /

n 5 >

n 2



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( 2

+ > 2(.Example 7:

3%1&

3%2&

$titte h = 2 in %2&

2 ( h = 16

2 ( = 16

2 ( + 2 = 16

= 16

= 16 +

= 20

4

2 + h =

9- in 1te;ati!n 3

h = = 20

= /

2 + h = < 2 h =16

2 + h =

2 h = 16

" > 10" 4



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E)er*!se1. e titti!n meth!d t! !lve the

f!ll!"in imltane! e;ati!n.

11 linear equations

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