Question

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

2x – 3y = 17, 4x + y = 13

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

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

2x – 3y = 17, 4x + y = 13

Answer 1

The given system of equations is:

2x – 3y – 17 = 0   ...(i)

4x + y – 13 = 0   ...(ii)

The given equations are of the form

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

where, a₁ = 2, b₁ = –3, c₁ = –17 and a₂ = 4, b₂ = 1, c₂ = –13

Now, `(a_1)/(a_2) = 2/4 = 1/2` and `(b_1)/(b_2) = (−3)/1 = -3`

Since, `(a_1)/(a_2) ≠ (b_1)/(b_2)`, therefore the system of equations has unique solution.

Using cross multiplication method, we have

`x/(b_1c_2 - b_2c_1) = y/(c_1a_2 - c_2a_1) = 1/(a_1b_2 - a_2b_1)`

⇒ `x/(-3(-13) − 1 xx (-17)) = y/(-17 xx 4 - (-13) xx 2) = 1/(2 xx 1 - 4 xx (-3))`

⇒ `x/(39 + 17) = y/(-68 + 26) = 1/(2 + 12)`

⇒ `x/56 = y/(-42) = 1/14`

⇒ `x = 56/14, y = (-42)/14`

⇒ x = 4, y = –3

Hence, x = 4 and y = –3.