Simultaneous Equations Example 1 & 2


Example 1:
Solve the simultaneous equations.
x+ 1 4 y=1 and  y 2 8=4x.

Solution:
x+ 1 4 y=1(1) y 2 8=4x(2) x=1 1 4 y(3)

Substitute (3) into (2),
y 2 8=4( 1 1 4 y ) y 2 8=4 4 4 y
y2 + y – 12 = 0
(y + 4)(y – 3) = 0
y = –4 or y = 3 

Substitute the values of y into (3),
when y=4,  x=1 1 4 (4)=5 when y=3,  x=1 1 4 (3)= 1 4 The solutions are x=5, y=4 and x= 1 4 , y=3


Example 2:
Solve the simultaneous equations 2x + y = 1 and 2x2 + y2 + xy = 5.
Correct your answer to three decimal places.

Solution:
2x + y = 1-----(1)
2x2 + y2 + xy = 5-----(2)

From (1),
y = 1 – 2x-----(3)

Substitute (3) into (2).
2x2 + (1 – 2x) 2 + x(1 – 2x) = 5
2x2 + (1 – 2x)(1 – 2x) + x – 2x2 = 5
1 – 2x – 2x + 4x2 + x – 5 = 0
4x2 – 3x – 4 = 0

From x= b± b 2 4ac 2a a=4, b=3c=4 x= ( 3 )± ( 3 ) 2 4( 4 )( 4 ) 2( 4 ) x= 3± 73 8 x=0.693 or 1.443

Substitute the values of x into (3).
When x = –0.693,
y = 1 – 2 (–0.693) = 2.386 (correct to 3 decimal places)

When x = 1.443,
y = 1 – 2 (1.443) = –1.886 (correct to 3 decimal places)

The solutions are x = –0.693, y  = 2.386 and x = 1.443, y = –1.886.