Question

Solve each of the following systems of equations by the method of cross-multiplication :

`ax + by = (a + b)/2`

3x + 5y = 4

Answer 1

The given system of equation is

`ax + by = (a + b)/2` .....(i)

3x + 5y = 4  ....(ii)

From (i), we get

2(ax + by) = a + b

`=> 2ax + 2by - (a + b) = 0`  .....(iii)

From (ii), we get

3x + 5y - 4 = 09

here

`a_1 = 2a, b_1 = 2b, c_1 = -(a + b)`

`a_2 = 3, b_2 = 5, c_2 = -4`

By cross multiplication, we hav-04e

`=> x/(2b xx (-4)- [-(a + b)]xx5) = (-y)/(2a xx (-4) - [-(a + b)] xx 3) = 1/(2a xx 5 - 2b xx 3) `

`=> x/(-8b + 5(a + b)) = (-y)/(-8a + 3(a + b)) = 1/(10a - 6b)`

`=> x/(-8b + 5a + 5b) = (-y)/(-8a + 3a + 3b) = 1/(10a - 6b)` 

`=> x/(5a - 3b) = (-y)/(-5a + 3b) = 1/(10a - 6b)`

Now

`x/(5a - 3b) = (-y)/(-5a + 3b) = 1/(10a - 6b)`

`=> x = (5a -3b)/(10a - 6b) = (5a - 3b)/(2(5a - 3b)) = 1/2`

And

`(-y)/(-5a + 3b) = 1/(10a - 6b)`

`=> -y = (-5a + 3b)/(2(5a - 3b))`

`=> y = (-(-5a + 3b))/(2(5a - 3b))`

`= (5a - 3b)/(2(5a - 3b))`

`=> y = 1/2`

Hence x = 1/2, y = 1/2  is the solution of the given system of equations