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.

Show that the following system of equations has a unique solution and solve it:

3x + 5y = 12, 5x + 3y = 4

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:

`(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)

⇒ 5y = 15

⇒ y = 3

Hence, x = –1 and y = 3 is the required solution.