Simultaneous equations

1
Simultaneous equations
Solve y 2x2 3x 4 and y 4x 1
2x2 3x 4 4x 1
? 2x2 7x 3 0
? (2x 1)(x 3) 0
? x ½ or x 3
y( ½ ) 4 ? ½ 1 3
y( 3 ) 4 ? 3 1 13
Complete solution ( ½ , 3) and (3, 13)
2
Simultaneous equations
Solve y x2 3x - 18 and y 2x 2
x2 3x - 18 2x 2
? x2 x - 20 0
? (x 5)(x - 4) 0
? x -5 or x 4
y( -5 ) 2? (-5) 2 -8
y( 4 ) 2 ? 4 2 10
Complete solution ( -5 , -8) and (4, 10)
3
Simultaneous equations
Show that the simultaneous equations y x2 3x
- 18 and y 2x 20 have no real solution.
x2 3x - 18 2x - 20
? x2 x 2 0
a 1, b 1 and c 2
? b2 4ac 1 4 ? 1 ? 2 - 7 lt 0
The line and the curve do not intersect i.e. no
real solutions
4
Simultaneous equations
The graph of y 2x k meets the graph of y
x2 3x 18 at only one point. Find the value of
k.
x2 3x - 18 2x k
? x2 x - k - 18 0
a 1, b 1 and c -k - 18
? b2 4ac 1 4 ? 1 ? (- k 18) 0
? 1 4k 72 0
? k -73/4
5
Simultaneous equations
It is given that x and y satisfy the simultaneous
equations y x 4 and 2x2 xy y2 8
(i) Show that x2 3x 2 0 (ii) Solve the
equations.
(i)
y x 4 ? y x 4
? 2x2 xy y2 8
? 2x2 x(x 4) (x 4)2 8
? 2x2 x2 4x x2 8x 16 8
? 4x2 12x 8 0
? x2 3x 2 0
? (x 2)(x 1) 0
(ii)
? x - 2 or x - 1
Complete solution (-2, 2) or (-1, 3)
6
Simultaneous equations
Solve the simultaneous equations. x y 1 and
x2 - xy y2 7
x y 1 ? y 1 - x
? x2 x(1 x) (1 x)2 7
? x2 x x2 1 2x x2 7
? 3x2 - 3x - 6 0
? x2 - x - 2 0
? (x 1)(x - 2) 0
? x - 1 or x 2
Complete solution (-1, 2) or (2, -1)
7
Solve the simultaneous equations x y 2
(1) x2 2y2 11 (2)
? Substitute for x from (1) into (2)
(2 y)2 2y2 11
4 4y y2 2y2 11
3y2 4y 7 0
? Factorise and solve for y
(3y 7)(y 1) 0
? Substitute for y in (1)
8
Solve the simultaneous equations y x2
(1) x y 6 (2)
? Substitute for x from (2) into (1)
y (6 y)2
y 36 12y y2
y2 13y 36 0
? Factorise and solve for y
(y 4)(y - 9) 0
y 4 or y 9
? Substitute for x in (2)