Question

Find the value of K for which each of
the following Sylfums of linean equations
has an infanited håmbers of Solutions.
20. (K-3) x + 3y =k,
kx + ky = 12

​

Answer 1

Given :-

• The pair of linear equations have infinite number of solutions.

To Find :-

• Value of k.

Solution :-

Given pair of linear equations are:-

• (k - 3)x + 3y = k
• kx + ky = 12

This can be rewritten as:-

• (k - 3)x + 3y - k = 0
• kx + ky - 12 = 0

Now, for infinite number of solutions, we have,

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

Putting the values for (a₁, a₂), (b₁, b₂), (c₁, c₂), we get,

$\implies\rm{\dfrac{k-3}{k}=\dfrac{3}{k}=\dfrac{-k}{-12}}$

Taking the first two terms, we get,

$\implies\rm{\dfrac{k-3}{k}=\dfrac{3}{k}}$

$\implies\rm{\dfrac{\cancel{k}(k-3)}{\cancel{k}}=3}$

$\implies\rm{k-3=3}$

$\implies\rm{k=6}$

Taking the last two terms, we get,

$\implies\rm{\dfrac{3}{k}=\dfrac{k}{12}}$

$\implies\rm{k^2=36}$

$\implies\rm{k=\sqrt{36}={\pm}6}$

Taking the common value, we get,

$\implies\rm{k=}\:\bold{6}$

Hence,

• The value of k = 6.