Question

8.Find the value of m and n for which the system of equations 3x + 4y = 12 and (m + n) x + 2 (m – n) y = 5m – 1 has infinite many solutions

Answer 1

$\large\underline{\sf{Given- }}$

Given pair of linear equation are

3x + 4y = 12

and

(m + n) x + 2 (m – n) y = 5m – 1

$\large\underline{\sf{To\:Find - }}$

m and n if pair of lines have infinitely many solutions.

$\large\underline{\sf{Solution-}}$

Given pair of linear equations are

3x + 4y = 12

and

(m + n) x + 2 (m – n) y = 5m – 1

We know, that

The pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 have

$\boxed{ \sf{ \:infinite \: solutions \: when   \rm \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}}}$

Now,

Comparing the given two equations with

a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, we get  

a₁ = 3

b₁ = 4

c₁ = 12

a₂ = m + n

b₂ = 2(m - n)

c₂ = 5m - 1

So, on substituting the values, we get

$\rm :\longmapsto\:\dfrac{3}{m + n} = \dfrac{4}{2(m - n)} = \dfrac{12}{5m - 1}$

can be rewritten as

$\rm :\longmapsto\:\dfrac{3}{m + n} = \dfrac{2}{m - n} = \dfrac{12}{5m - 1}$

Taking first and second member, we get

$\rm :\longmapsto\:\dfrac{3}{m + n} = \dfrac{2}{m - n}$

$\rm :\longmapsto\:3m - 3n = 2m + 2n$

$\bf\implies \:m = 5n - - - (1)$

Now, taking second and third member, we get

$\rm :\longmapsto\: \dfrac{2}{m - n} = \dfrac{12}{5m - 1}$

$\rm :\longmapsto\: \dfrac{1}{m - n} = \dfrac{6}{5m - 1}$

$\rm :\longmapsto\:6m - 6n = 5m - 1$

$\rm :\longmapsto\:m= 6n - 1$

$\rm :\longmapsto\:5n= 6n - 1$

$\rm :\longmapsto\:5n - 6n = - 1$

$\rm :\longmapsto\: -n = - 1$

$\bf :\longmapsto\: n = 1 - - - (2)$

On substituting n = 1, in equation (1), we get

$\bf\implies \:m = 5 \times 1 = 5$

Hence,

$\: \: \: \: \: \: \: \: \: \: \purple{\boxed{ \bf{ \:m \: = \: 5}} \: \: \rm \: and \: \: \: \boxed{ \bf{ \:n \: = \: 1}}}$

Additional Information :-

The pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 have

$\boxed{ \sf{ \:unique \: solution \: when \:   \rm \dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}}}$

$\boxed{ \sf{ \:no \: solution \: when \:   \rm \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}}}$