Question

Find the value of k for which each of the following systems of equation have infinitely many solution:
$2x + 3y = 2$
$\left( {k + 2} \right)y + \left( {2k + 1} \right)y = 2\left( {k - 1} \right)$

Answer 1

Hint – Here we will proceed by using the no solution condition for linear equation that is $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$. By applying this condition, we will get our answer.

Complete Step-by-Step solution:
If we have linear equation in the following form,
$
  {a_1}x + {b_1}y + {c_1} = 0 \\
  {a_2}x + {b_2}y + {c_1} = 0 \\
 $
For these type of equations, condition of no solution will be,
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$ (If this condition is satisfied then we can say that equations has no solution)
Given equation is
$2x + 3y = 2$…. (1)
We can also write it as,
$\left( {k + 2} \right)y + \left( {2k + 1} \right)y = 2\left( {k - 1} \right)$…. (2)
Now, by comparing equation (1) and (2) by $
  {a_1}x + {b_1}y + {c_1} = 0 \\
  {a_2}x + {b_2}y + {c_1} = 0 \\
 $ we will get,
For, equation (1)
${a_1} = 2,{b_1} = 3,{c_1} = - 2$
For, equation (2),
${a_2} = k + 2 + ,{b_2} = 2k + 1,{c_2} = - 2\left( {k - 1} \right)$
Now we will apply the no solution condition here that is $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$ we will get,
$\dfrac{2}{{k + 2}} = \dfrac{3}{{2k + 1}} \ne \dfrac{{ - 2}}{{ - 2\left( {k - 1} \right)}}$ …. (3)
Now, we will simplify the equation,
$\dfrac{2}{{k + 2}} = \dfrac{3}{{2k + 1}}$
$ \Rightarrow 2\left( {2k + 1} \right) = 3\left( {k + 2} \right)$ ….. (Transposing)
$ \Rightarrow 4k + 2 = 3k + 6$
$ \Rightarrow 4k - 3k = 6 - 2$
$ \Rightarrow k = 4$
So, we can say that, if the value of $k = 4$ then infinite solutions will exist for this system.

Note – Whenever we come up with this type of question, one must know that this question can be solved by various methods such as matrix method, we can also solve this method graphically. It is a consistent system of equations.