Question

Find the value of k for which each of the following systems of equations have infinitely many solution:
4x + 5y = 3
kx + 15y = 9

Answer 1

Hint: Here we will proceed by using the condition of infinite many solution i.e. $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$. Then we will compare the ratios of the coefficients of the given equation and get the required value of k.
Complete Step-by-Step solution:
As we are given a system of equations i.e. 4x + 5y = 3……. (1) and kx + 15y = 9………… (2)
The above equations are of the form-
$
  {a_1}x + {b_1}y - {c_1} = 0 \\
  {a_2}x + {b_2}y - {c_2} = 0 \\
 $
We will apply condition of infinite many solution-
i.e. $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$
So according to question,
Here ${a_1} = 4,{b_1} = 5,{c_1} = - 3$
And ${a_2} = k,{b_2} = 15,{c_2} = - 9$
Now applying above condition,
we get-
$ \Rightarrow \dfrac{4}{k} = \dfrac{5}{{15}} = \dfrac{{ - 3}}{{ - 9}}$
$\Rightarrow \dfrac{4}{k} = \dfrac{1}{3}$
$\Rightarrow k = 12$
Hence the given system of equations will have infinitely many solutions, if k = 12.

Note: We can also use another method to solve this question, firstly we convert the system of equations into matrix form in terms of A B and X. If there’s infinitely many solutions of the system of equations, then the value of adjoint$\left( A \right) \times B = 0$. Further we will use the method of multiplication of two matrices, we will find the value of adjoint$\left( A \right) \times B = 0$. Eventually we will compare the values and get the value of x and y.