Question

if x + 2y + 3z = 14,<br />2x + 3y + z = 11 and<br />3x + y + 2z = 11,<br />find (x + y + z)​

Answer 1

Answer:

Required value of x + y + z is 6.

Step-by-step-explanation:

Given,

x + 2y + 3z = 14 ... ( i )

2x + 3y + z = 11 ...( ii )

3x + y + 2z = 11 ... ( iii )

Comparing ( iii ) and ( ii ) :

= > 3x + y + 2z = 2x + 3y + z

= > 3x - 2x + 2z - z = 3y - y

= > x + z = 2y

In ( i ) :

= > x + 2y + 3z = 14

= > x + 2y + 2z + z = 14

= > x + z + 2y + 2z = 14

= > 2y + 2y + 2z = 14

= > y + y + z = 7

= > 2y + z = 7

In ( iii ) :

= > 3x + y + 2z = 11

= > 3x + y - z + z + 2z = 11

= > 3x + y - z + 3z = 11

= > 3x + 3z + y - z = 11

= > 3( x + z ) + y - z = 11

= > 3( 2y ) + y - z = 11

= > 6y + y - z = 11

= > 7y - z = 11

From above,

= > ( 2y + z ) + ( 7y - z ) = 7 + 11

= > 2y + 7y = 18

= > 9y = 18

= > y = 2

Substituting the value of y in ( i ) & ( ii ) :

= > x + 2y + 3z = 14 { In ( i ) }

= > x + 2( 2 ) + 3z = 14

= > x + 3z = 14 - 4

= > x + 3z = 10 ....( 4 )

= > 2x + 3y + z = 11

= > 2x + 3( 2 ) + z = 11

= > 2x + z = 11 - 6

= > 2x + z = 5

Multiply ( 4 ) by 2 and subtracting from this :

= > 2x + z - 2( x + 3z ) = 5 - 2( 10 )

= > 2x + z - 2x - 6z = 5 - 20

= > - 5z = - 15

= > z = 3

Therefore, x = 1

Hence,

= > x + y + z

= > 1 + 2 + 3

= > 6

Answer 2

Answer:

x + y + z = 6 .

Step-by-step explanation:

Given ;

x + 2 y + 3 z = 14

Multiplying by 2 we get

2 x + 4 y + 6 z = 28

2 x = 28 - 4 y - 6 z ... ( i )

2 x + 3 y + z = 11   [ Given ]

2 x = 11 - 3 y - z ...  ( ii )

From ( i )  and ( ii )

28 - 4 y - 6 z = 11 - 3 y - z

y + 5 z = 17 ... ( iii )

Again multiply by 3 in ( x + 2 y + 3 z = 14 ) we get

3 x + 6 y + 9 z = 42

3 x = 42 - 6 y - 9 y ... ( iv )

3 x + y + 2 z = 11  [ Given ]

3 x = 11 - y - 2 z ... ( v )

From  ( iv )  and  ( v )  we have

42 - 6 y - 9 y = 11 - y - 2 z

5 y + 7 z = 31 .

Now we have ;

y + 5 z = 17  & 5 y + 7 z = 31

Multiply by 5 in first

5 y = 85 - 25 z   & 5 y = 31 - 7 z

Comparing both

85 - 25 z = 31 - 7 z

18 z = 54

z = 3

Putting z = 2 in y + 5 z = 17

y = 17 - 15

y = 2

Now putting y and z value in (  x + 2 y + 3 z = 14 )

x = 14 - 9 - 4

x = 1

Now ;

x + y + z = 1 + 2 + 3

x + y + z =  6 .

Thus we get answer 6 .