Simultaneous Equations Long Questions (Question 3 & 4)


Question 3:
Solve the following simultaneous equations.
3y – 2x= – 4
y2 + 4x2 = 2

Solution:
3y – 2x= – 4 —–(1)
y2 + 4x2= 2 —–(2)
From (1), y = 2 x 4 3  ——- ( 3 ) Substitute (3) into (2), ( 2 x 4 3 ) 2 + 4 x 2 = 2 ( 4 x 2 16 x + 16 9 ) + 4 x 2 = 2 4 x 2 16 x + 16 + 36 x 2 = 18     ( × 9 ) 40 x 2 16 x 2 = 0 20 x 2 8 x 1 = 0 ( 10 x + 1 ) ( 2 x 1 ) = 0 x = 1 10   or   x = 1 2 Substitute the values of  x  into (3), When  x = 1 10 , y = 2 ( 1 10 ) 4 3 = 1 2 5 When  x = 1 2 , y = 2 ( 1 2 ) 4 3 = 3 3 = 1 The solutions are  x = 1 10 ,   y = 1 2 5  and  x = 1 2 ,   y = 1.



Question 4:
Solve the simultaneous equations x – 3y = –1 and y + yx– 2x = 0.
Give your answers correct to three decimal places.

Solution:
x – 3y = –1 —–(1)
y + yx – 2x = 0 —–(2)
From (1),
x = 3y – 1 —–(3)
Substitute (3) into (2),
y + y (3y – 1) – 2(3y – 1)  = 0
y + 3y2y – 6y+ 2 = 0
3y2 – 6y + 2 = 0

a = 3 ,   b = 6 c = 2 y = b ± b 2 4 a c 2 a y = ( 6 ) ± ( 6 ) 2 4 ( 3 ) ( 2 ) 2 ( 3 ) y = 6 ± 12 6 y = 1.577  or 0 .423

Substitute the values of y into (3).
When y = 1.577,
x = 3 (1.577) – 1 = 3.731 (correct to 3 decimal places)

When y = 0.423,
x = 3 (0.423) – 1 = 0.269 (correct to 3 decimal places)

The solutions are x = 3.731, y = 1.577 and x = 0.269, y = 0.423.


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