Question

for what values of a and b the following system of linear equation has infinite no of solution (a-1)x-3y=5,3x+(b-2)y=3​

Answer 1

Step-by-step explanation:

the value of a is -6 and the value of b is 1/5

becoz for the linear equation having infinite no. of solutions the system is a1/a2 = b1/b2 = C1/c2

thanks:))

Answer 2

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

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

Given pair of lines are

• (a - 1)x - 3y = 5

• 3x + (b-2)y = 3 have infinitely many solutions.

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

• The values of a and b.

Calculation :-

Given that pair of lines are

• (a - 1)x - 3y = 5

• 3x + (b-2)y = 3 have infinitely many solutions.

$\sf \: Let \: consider \: lines \: a_1x + b_1y + c_1 = 0 \: and \: a_2x + b_2y + c_2 = 0$

then,

System of equations have infinitely many solutions iff

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

Here,

• a₁ = a - 1

• a₂ = 3

• b₁ = - 3

• b₂ = b - 2

• c₁ = 5

• c₂ = 3

Now,

On substituting the values, we get

$\rm :\longmapsto\:\dfrac{a - 1}{3} = \dfrac{ - 3}{b - 2} = \dfrac{5}{3}$

☆ On taking first and third member, we get

$\rm :\longmapsto\:\dfrac{a - 1}{3} = \dfrac{5}{3}$

$\rm :\longmapsto\:a - 1 = 5$

$\bf\implies \:a = 6$

Now,

☆ On taking second and third member, we get

$\rm :\longmapsto\: \dfrac{ - 3}{b - 2} = \dfrac{5}{3}$

$\rm :\longmapsto\: - 9 = 5b - 10$

$\rm :\longmapsto\: 10- 9 = 5b$

$\rm :\longmapsto\: 1= 5b$

$\bf\implies \:b = \dfrac{1}{5}$

Additional Information :-

$\sf \: Let \: consider \: lines \: a_1x + b_1y + c_1 = 0 \: and \: a_2x + b_2y + c_2 = 0$

then

1. System of lines have no solution iff

$\bf \:\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$

2. System of lines have unique solution iff

$\bf \:\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$