Simultaneous Equations, Long Questions


Simultaneous Equations, Long Questions 
Question 7:
Solve the following simultaneous equations.
5y – 6x= 2
4 y x 3 x y = 4.

Solution:
5y – 6x= 2 —– (1)
4 y x 3 x y = 4  ------- (2) From (1), y = 2 + 6 x 5

Substitute (3) into (2),
4 ( 2 + 6 x 5 ) x 3 x ( 2 + 6 x 5 ) = 4 8 + 24 x 5 x 15 x 2 + 6 x = 4 ( 8 + 24 x ) ( 2 + 6 x ) ( 15 x ) ( 5 x ) 5 x ( 2 + 6 x ) = 4

16 + 48x + 48x + 144x2 – 75x2 = 20x (2 + 6x)
69x2 + 96x + 16 = 40x + 120x2
51x2 – 56x – 16 = 0
(3x – 4)(17x + 4) = 0
3x – 4 = 0 or 17x + 4 = 0
x = 4 3   or    x = 4 17
  

Substitute  x = 4 3  into (3), y = 2 + 6 ( 4 3 ) 5 = 2

Substitute  x = 4 17  into (3), y = 2 + 6 ( 4 17 ) 5 = 2 17

The solutions are  ( 4 3 , 2 )  and  ( 4 17 , 2 17 ) .


Question 8:
Solve the following simultaneous equations.
x+2y=1 2 x 2 + y 2 +xy=5
Give your answer correct to three decimal places.

Solution:
x+2y=1..........( 1 ) 2 x 2 + y 2 +xy=5..........( 2 ) x=12y..........( 3 ) Substitute ( 3 ) into ( 2 ), 2 ( 12y ) 2 + y 2 +( 12y )y=5 2( 14y+4 y 2 )+ y 2 +y2 y 2 =5 28y+8 y 2 y 2 +y5=0 7 y 2 7y3=0 a=7,b=7,c=3 x= b± b 2 4ac 2a y= ( 7 )± ( 7 ) 2 4( 7 )( 3 ) 2( 7 ) y= 7± 133 14 y=1.324     or     0.324 From x=12y When y=1.324,  x=12( 1.324 )=1.648 When y=0.324,  x=12( 0.324 )=1.648 The solutions are ( 1.648,1.324 ) and ( 1.648,0.324 ).

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