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:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
where, a₁ = 3, b₁= 5, c₁ = −12 and a₂ = 5, b₂ = 3, c₂ = −4
For a unique solution, we must have:
$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$, i.e., $\frac{3}{5}\ne \frac{5}{3}$
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.