Question

If am ≠ bl, then the system of equations

ax + by = c

lx + my = n

A. has a unique solution

B. has no solution

C. has infinitely many solutions

D. may or may not have a solution.

Answer 1

Given:

Equation 1: ax + by = c

Equation 2: lx + my = n

Both the equations are in the form of :

a₁x + b₁y = c₁ & a₂x + b₂y = c₂ where

a₁ & a₂ are the coefficients of x

b₁ & b₂ are the coefficients of y

c₁ & c₂ are the constants

According to the problem:

a₁ = a

a₂ = l

b₁ = b

b₂ = m

c₁ = c

c₂ = n

According to the question the condition given is am ≠ bl …(i)

To develop a relationship between the coefficients we divide both sides of the equation by l*m

After dividing we get

Since

a₁ = a,

a₂ = l,

b₁ = b,

b₂ = m

So

We know from our properties of linear equations that if the ratio of the coefficients of x and y are not equal then their exists a unique solution.

The given system of equation has a unique solution for all values of x and y