Question

Show that the following system of equations has a unique solution:
3x + 5y = 12,
5x + 3y = 4.
Also, find the solution of the given system of equations.

Answer 1

The given system of equations is:

3x + 5y = 12
5x + 3y = 4
These equations are of the forms:
`a_1x+b_1y+c_1 = 0 and a_2x+b_2y+c_2 = 0`
where, `a_1 = 3, b_1= 5, c_1= -12 and a_2 = 5, b_2 = 3, c_2= -4`
For a unique solution, we must have:
`(a_1)/(a_2) ≠ (b_1)/(b_2), i.e., 3/5 ≠ 5/3`
Hence, the given system of equations has a unique solution.
Again, the given equations are:
3x + 5y = 12 …..(i)
5x + 3y = 4 …..(ii)
On multiplying (i) by 3 and (ii) by 5, we get:
9x + 15y = 36 …….(iii)
25x + 15y = 20 ……(iv)
On subtracting (iii) from (iv), we get:
16x = -16
⇒x = -1
On substituting x = -1 in (i), we get:
3(-1) + 5y = 12
⇒5y = (12 + 3) = 15
⇒y = 3
Hence, x = -1 and y = 3 is the required solution.