Question

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

`x/3 + y/2 = 3, x - 2y = 2`

Also, find the solution of the given system of equations.

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

`x/3 + y/2 = 3, x - 2y = 2`

Answer 1

The given system of equations is:

`x/3 + y/2 = 3`

⇒ `(2x + 3y)/6 = 3`

2x + 3y = 18

⇒ 2x + 3y – 18 = 0   ...(i)

and

x – 2y = 2

x – 2y – 2 = 0   ...(ii)

These equations are of the forms:

a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

where, a₁ = 2, b₁ = 3, c₁ = –18 and a₂ = 1, b₂ = –2, c₂ = –2

For a unique solution, we must have:

`(a_1)/(a^2) ≠ (b_1)/(b_2)`, i.e., `2/1 ≠ 3/(−2)`

Hence, the given system of equations has a unique solution.

Again, the given equations are:

2x + 3y – 18 = 0   ...(iii)

x – 2y – 2 = 0   ...(iv)

On multiplying (i) by 2 and (ii) by 3, we get:

4x + 6y – 36 = 0   ...(v)

3x – 6y – 6 = 0   ...(vi)

On adding (v) from (vi), we get:

7x = 42

⇒ x = 6

On substituting x = 6 in (iii), we get:

2(6) + 3y = 18

⇒ 3y = (18 – 12)

⇒ 3y = 6

⇒ y = 2

Hence, x = 6 and y = 2 is the required solution.