Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.51

2.7 SIMULTANEOUS LINEAR EQUATIONS

Introduction :
A linear equation of two unknowns x and y, is of the form ax + by + c = 0, where a > 0, b > 0.
Two such equations : a 1 x + b 1 y + c 1 = 0
a 2 x + b 2 y + c 2 = 0
form two simultaneous linear equations in x and y.
Methods of Solution
There are two methods, such as–
(1) Method of elimination,
(2) Rule of cross-multiplication
Method of Elimination
Example 85 :
Solve 3x + 4y = 11 ... (i) 5x – 2y = 1 ... (ii)
Solution :
Multiplying eqn. (I) by 5 and eqn. (ii) by 3, we find
15x + 20y = 55 .... (iii)
15x – 6y = 3 ..... (iv)
Now subtracting eqn. (iv) from eqn. (iii), 26y = 52 or, y = 2.
Putting this value of y in eqn. (i), 3x + 4.2 = 11 or, 3x = 11 – 8 or, x = 1
∴ x = 1, y = 2.
Rule of Cross-multiplication :
If two relations amongst three unknowns are given, the ratios of the three unknowns can be obtained by
the rule of cross-multiplication.
For example, let such two relations are given by the following equations :
a 1 x + b 1 y + c 1 z = 0 .... (1)
a 2 x + b 2 y + c 2 z = 0 .... (2)
The rule states

1 2 1 2 1 2 1 2 1 2 1 2

x y z

b c cb c a a c ab b a− = − = −

Example 86 : Solve 3x + 4y = 11 .... (1) 5x – 2y = 1 .... (2)
Solution :
The expression can be written as
3x + 4y – 11 = 0, a 1 = 3, b 1 = 4, c 1 = – 11
5x – 2y – 1 = 0, a 2 = 5, b 2 = – 2, c 2 = – 1

( ) ( )( )

( )

x =4. -1 - -11. -2 =-4 -22 -26= =1

∴ 3 -2 - 4.5 -6- 20 -26

( ) ( )

( )

y 11 5 3 1 55 3 52 2.

3 2 4.5 6 20 26

=− − − =− + =− =

− − − − −