Question

the pair of equations 3x+y=81,81x+y =3 has no solution, unique solution, infinite many solution

Answer 1

3x + y = 81
81x + y = 3

a1 3. b1 1. c1 81
a2 81. b2 1 c2 3

a1. b1
a2 is not equal to b2

so this equation have unique solution

hope this helps ☺☺

Answer 2

The pair of equations 3x+y=81 and 81x+y=3 has a unique solution.

Solution: For a pair of linear equations ax+by+c=0 and px+qy+r=0, the necessary conditions for:-

1- having a unique solution

$\frac{a}{p} ≠ \frac{b}{r}$

2- having no solution

$\frac{a}{p} = \frac{b}{q} ≠ \frac{c}{r}$

3- for infinite many solutions

$\frac{a}{p} = \frac{b}{q} = \frac{c}{r}$

Here, 3x+y= 81 can be written as 3x+y-81=0

So, a= 3, b= 1 and c= -81

81x+y=3 can be written as 81x+y-3=0

So, p= 81, q=1 and r= -3

Here,

a/p= 3/81= 1/27

b/q= 1/1= 1

c/r= -81/-3= 27

Clearly, a/p ≠ b/q which is a necessary condition for a unique solution for a pair of linear equations.