Question

The following determinants are obtained from the simultaneous equations in variable x and y.

If D_x = `|(24, a),(16, -1)|`, D_y = `|(5, 24),(b, 16)|`, D = `|(5, 1),(3, -1)|` the solution for this equations are x = 5 and y = – 1. Find the value of' a' and 'b'. Also form the original simultaneous equations having this solution.

Answer 1

Given: D_x = `|(24, a),(16, -1)|`, D_y = `|(5, 24),(b, 16)|`, D = `|(5, 1),(3, -1)|` 

D_x = `|(24, a),(16, -1)| = - 24 - 16a`

D_y = `|(5, 24),(b, 16)| = 80 - 24b`

D = `|(5, 1),(3, -1)| = - 5 - 3 = - 8` 

According to Cramer's rule,

x = `D_x/D`

⇒ 5 = `(-24 -  16a)/(-8)`

⇒ 5 = 3 + 2a

⇒ 5 – 3 = 2a

⇒ a = 1

Also, y = `D_y/D`

⇒ – 1 = `(80 - 24b)/(-8)`

⇒ – 1 = – 10 + 3b

⇒ – 1 + 10 = 3b

⇒ b = 3

So, a = 1 and b = 3.

So, D_x = `|(24, 1),(16, -1)|` and D_y = `|(5, 24),(3, 16)|`

As a result, the solutions to the original simultaneous equations are 5x + y = 24 and 3x – y = 16.