1. SIMULTANEOUS EQUATIONS
(i) Characteristics of simultaneous equations: example
(a) Involves TWO variables, usually in x and y. 4x + y = –8
(b) Involves TWO equations : one linear and x2 + x – y = 2
the other non- linear.
(ii) “ Solving simultaneous equations” means finding the values of x
and corresponding y which satisfy BOTH the equations.
BASIC SKILLS REQUIRED
SKILL 1 SKILL 2 SKILL 3
Changing the subject of Expansion of Algebraic equation to get Solving Quadratic Equations
the formula quadratic equation ax2+bx+c=0 ax2+bx + c = 0
examples examples examples
1 3x + y = 6 y= 1 (2 + x)2 =1 7 3x(– 3 – 2x = 0 1 By factorization
Solve
2x – y = 3 y= 2 (4x - 5)2 =2 8 (x – 4 )(2x) =x+2 x2 – 3x – 10= 0
2 (x + 2)(x – 5) = 0
x = -2, x = 5
3 x
+ 3y = 9 x = 3 (3 – 2x)2=0 9 (x + 1)(2x – 3) =
2
2 2 By using formula
4 2x – 3y = 2 x = 4  1 − 2x  10 (2x – 3 )(2x + 3) =

 3 
 =0 − b ± b2 − 4ac
x=
7x – 2y = 5 x = 5 2 11 2 1 2a
5  5 + 3x 
  = + = Solve 2x2 – 8x + 7 = 0
3 
 3x 3 − 3x
y= − ( −8 ) ± ( −8 )2 − 4( 2 )( 7 )
6 x y
+ =1 6  3x − 4 
2 12 2 x=
  =  1 − 3x   − 3x  2( 2 )
2 3  2  3  + 4x  
 2   2  = 2.707 or 1.293
Method of Solving Simultenous Solve Solve
Equations 4x + y = -8 and x2 + x – p - m = 2 and p2 + 2m = 8
y=2
1) Starting from the LINEAR
equation, express y in terms of x y = – 8 -4x m = p -2
(or x in terms of y).
2) Substitute y (or x) into the x2 + x – y = 2 p2 + 2m = 8
second equation (which is non- x2 + x – (-8-4x) = 2
p2 + 2 ( p – 2) = 8
linear) to obtain a quadratic 2
x + x + 8+ 4x = 2
equation in the form x2 + 5x + 6 = 0 p2 + 2p - 4 = 8
2
ax + bx + c = 0. p2 + 2p -12 = 0
3) Solve the quadratic equation (x + 2) (x + 3) = 0 2
by factorisation or by using the p = − 2 ± 2 − 4( 1 )( −12 )
x = -2 , x = -3 2
− b ± b 2 − 4ac = 2.606 , - 4.606
FORMULA x= .
2a
4) Find the If x = -2, y = – 8 -4(-2) If p = 2.606 , m = 0.606
corresponding value of x or y. = 0
If x = -3, y = – 8 -4(-3) If p = - 4.606, m = -6.606
= 4