Question

in two linear equation a1x+b1y +c1 =0 and a2x +b2y +c2=0,if a1/a2=b1/b2 =c1/c2 then write the number of solutions these pair of equations have​

Answer 1

Answer:-

• 1st equation = a₁x + b₁y + c₁ = 0

• 2nd equation = a₂x + b₂y + c₂ = 0

⠀⠀⠀⠀⠀⠀⠀⠀According to the Question

It is given that ,

$:\implies$ a₁/a₂ = b₁/b₂ = c₁/c₂

So, the given equation has infinitely many solution .

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Additional Information !!

a₁x + b₁y + c₁ and a₂x + b₂y + c₂ are two linear equations . If

$\longrightarrow$ a₁/a₂ ≠ b₁/b₂

In this case there is only one solution.

And ,

$\longrightarrow$ a₁/a₂ = b₁/b₂ ≠ c₁/c₂

In this case there is no solution .

Answer 2

Given : The two linear equations are : $\sf a_1x \:+ \: b_1 y \: + \:c_1 \:\: =\:0 \: [ \: as \:,\: Eq^n \: 1 \: \:] \: \bf , \:\sf a_2x \:+ \: b_2 y \: + \:c_2 \:\: =\:0 \: [ \: as \:,\: Eq^n \: 2 \: \:] \: \bf \& \:\sf \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$

Exigency To Find : The number of the solutions the equation have ?

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

⠀⠀⠀⠀⠀Given that ,

$\qquad \leadsto \sf \: a_1x \:+ \: b_1 y \: + \:c_1 \:\: =\:0 \: \qquad \:\:\bigg\lgroup \sf{ Equation \: 1 \:}\bigg\rgroup$

$\qquad \leadsto \sf \: a_2x \:+ \: b_2y \: + \:c_2\:\: =\:0 \: \qquad \:\:\bigg\lgroup \sf{ Equation \: 2 \:}\bigg\rgroup$

⠀⠀⠀⠀⠀Here ,

$\qquad \qquad \star \:\: \sf \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$

⠀⠀⠀⠀⠀As , We know that ,

⠀⠀▪︎ ⠀⠀ If $\sf a_1x \:+ \: b_1 y \: + \:c_1 \:\: =\:0 \: [ \: as \:,\: Eq^n \: 1 \: \:] \: \bf , \:\sf a_2x \:+ \: b_2y \: + \:c_2 \:\: =\:0 \: [ \: as \:,\: Eq^n \: 2 \: \:] \:\:$ are two linear Equations and if both Equations have :

$\qquad \leadsto \:\: \sf \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$

• The Equation have infinite number of Solutions .

$\qquad \leadsto \:\: \sf \dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$

• The Equation have only one Solution .

$\qquad \leadsto \:\: \sf \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq\dfrac{c_1}{c_2}$

• The Equation have no solution .

Here In the given Equation :

$\qquad \qquad \star \:\: \sf \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$

Then ,

• The Equation have infinite number of solutions .