Question

Find the value of $k$ for which each of the following system of equations have infinitely many solution:
$
  2x + \left( {k - 2} \right)y = k \\
  6x + \left( {2x - 1} \right)y = 2k + 5 \\
 $

Answer 1

Hint – We will start solving this question by writing down the given equations. By using the given condition, i.e., equations have infinitely many solutions, we will use an appropriate formula i.e. $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$ which will give the required value.

Complete Step-by-Step solution:
Here, we have the equations in the form of,
$
  {a_1}x + {b_1}y + {c_1} = 0 \\
  {a_2}x + {b_2}y + {c_2} = 0 \\
 $ ……………. (1)
And the equations have infinitely many solution, then the formula to be used is,
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$ …………… (2)
Now, the given equations are,
$
  2x + \left( {k - 2} \right)y = k \\
  6x + \left( {2x - 1} \right)y = 2k + 5 \\
 $
Comparing with (1), we have
$
  {a_1} = 2,{b_1} = \left( {k - 2} \right),{c_1} = - k \\
  {a_2} = 6,{b_2} = \left( {2k - 1} \right),{c_2} = - \left( {2k + 5} \right) \\
 $
Now, putting these values in (2), we obtain
$\dfrac{2}{6} = \dfrac{{\left( {k - 2} \right)}}{{\left( {2k - 1} \right)}} = \dfrac{{ - k}}{{ - \left( {2k + 5} \right)}}$
$ \Rightarrow \dfrac{1}{3} = \dfrac{{k - 2}}{{2k - 1}} = \dfrac{k}{{2k + 5}}$
$ \Rightarrow \dfrac{1}{3} = \dfrac{{k - 2}}{{2k - 1}}$ ……. (3) and $\dfrac{1}{3} = \dfrac{k}{{2k + 5}}$ ……. (4)
First we will solve equation (3),
$
  \dfrac{1}{3} = \dfrac{{k - 2}}{{2k - 1}} \\
   \Rightarrow 2k - 1 = 3\left( {k - 2} \right) \\
   \Rightarrow 2k - 1 = 3k - 6 \\
   \Rightarrow 2k - 3k = - 6 + 1 \\
   \Rightarrow - k = - 5 \\
   \Rightarrow k = 5 \\
 $
Now, we will solve equation (4),
$
  \dfrac{1}{3} = \dfrac{k}{{2k + 5}} \\
   \Rightarrow 2k + 5 = 3k \\
   \Rightarrow 2k - 3k = - 5 \\
   \Rightarrow - k = - 5 \\
   \Rightarrow k = 5 \\
 $
Hence, the value of $k$ is 5.

Note – A linear equation is an equation in which the highest degree term in the variable or variables is of the first degree and when a graph is plotted gives a straight line. While using this formula make sure that you do not get confused between the formulas of equations having different types of solutions.