Question

Solve the following system of equation by the method of Cross Multiplication. 2x + y = 35; 3x + 2y = 65

Answer 1

Answer:

x=15 and y=5

Step-by-step explanation:

Concept :

CROSS - MULTIPLICATION METHOD:

The general form of a pair of linear equations

a1x + b1y + c1 = 0 , &        a2x + b2y + c2 = 0.

When a1 / a2 ≠ b1 / b2, the pair of linear equations will have a unique solution.

To solve this pair of equations for x and y using cross-multiplication, we’ll arrange the variables x and y and their coefficients a1, a2, b1 and b2, and the constants c1 and c2  as shown below

⇒ x =  b1 c2 - b2 c1  / a1 b2 - a2 b1    

⇒ y =  c1 a2 - c2 a1  / a1 b2 - a2 b1

The above equation is generally written as :

x/ b1 c2 - b2 c1 =  y/ c1 a2 - c2 a1 = 1/a1 b2 - a2 b1

x                       y                        1

-----------   =   -----------------     =    ---------

b1      c1        c1           a1           a1        b1

b2      c2       c2           a2           a2       b2

Given :

2x + y = 35

3x + 4y = 65

 

Solution :  

We have ,  

2x + y - 35 = 0

3x + 4y - 65 = 0

Here a1 = 2, b1 = 1, c1 = - 35

a2 = 3, b2 = 4, c2 = - 65

x                       y                        1

-----------   =   -----------------     =    ---------

1      -35         - 35         2           2       1

4    -65           - 65       3            3     4

x/( 1 × - 65) - (4 × - 35) = y/(- 35 × 3) - (- 65 × 2) = 1 /(2 × 4) - (3 × 1)

x/- 65 + 140 = y/- 105 + 130 = 1/ 8 - 3

x/75 = y/25 = 1/5

Now, x/75  = 1/5

x =  75/5

x = 15

And  

y/25 = 1/5

y = 25/5

 y = 5

Hence the value of given systems of equations is x = 15 and y = 5