Question

For what value of p , the following pair of linear equations in two variables will have infinitely many solutions ? PX + 3Y= 0 ,12x + py - p = 0

Answer 1

we know the condition of infinitely many solutions ,
if pair of linear equations in two variables are
$a_1x+b_1y+c_1=0\\a_2x+b_2y+c_2=0$
then, $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ is the condition of infinitely many solutions.

so, P/12 = 3/P = 0/P [ something is wrong in question]

so, you just above expression to solve in case of infinitely many solutions.

Answer 2

Given:

Linear equation 1 =  Px + 3y - ( p - 3 ) = 0

Linear equation 2 = 12x +py - p = 0

To find:

The value of p, the following pair of linear equation in two variables will have infinitely many solutions.

Solution:

Condition for infinitely many solutions,

a1 / a2 = b1 / b2 = c1 / c2

Here,

a1= p  

b1 = 3

c1 = - ( p - 3 )

a2 = 12

b2 = p

c2 = -p

Substituing,

p / 12 = 3 / p = - ( p - 3 ) / -p  

p / 12 = 3 / p

Solving,

p^2 = 36

Hence,

p = ± 6  

If p = -6, the condition cannot be satisfied.

Hence,

p = 6

Hence, the pair of linear equation in two variables will have infinitely many solutions if p = 6.

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