Question

find the value of 'k'
3x+ y =7
4x+ky =9 has a unique solution​

Answer 1

$\sf{\bold{\blue{\underline{\underline{Given}}}}}$

● pair of equation

• 3x+y=7

• 4x+ky=9

⠀⠀⠀⠀

$\sf{\bold{\red{\underline{\underline{To\:Find}}}}}$

■ Value of "k".??■⠀⠀⠀⠀

$\sf{\bold{\purple{\underline{\underline{Solution}}}}}$

➔ Here it is given that the pair of equations have a unique solution

➔ If a pair of equations have a unique solution we know that,

    $\tt{\dfrac{a_1}{a_2}\neq \dfrac{b_1}{b_2}}$

where,

a1=3

a2=4

b1=1

b2=k

so..

$\rightsquigarrow{\dfrac{a_1}{a_2}\neq \dfrac{b_1}{b_2}}$

$\rightsquigarrow{\dfrac{3}{4}\neq \dfrac{1}{k}}$

$\rightsquigarrow{3×k}\neq {1×4}$

$\rightsquigarrow{3k}\neq {4}$

$\rightsquigarrow{k}\neq \dfrac{4}{3}$

$\sf{\bold{\green{\underline{\underline{Answer}}}}}$

$\rightsquigarrow{\boxed{\orange{k\neq{\frac{4}{3}}}}}$⠀⠀

$\sf{\bold{\blue{\underline{\underline{Notes:-}}}}}$

$\rightsquigarrow$If a pair of equations:

    a₁x + b₁y + c₁ = 0

    a₂x + b₂y + c₂ = 0

$\rightsquigarrow$has a unique solution and is consistent,

$\red{\dfrac{a_1}{a_2}\neq \dfrac{b_1}{b_2}}$

$\rightsquigarrow$Has infinite number of solutions and is consistent,

$\red{\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2} }$

$\rightsquigarrow$Has no solution and is inconsistent,

$\red{\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq \dfrac{c_1}{c_2} }$