Question

Find the value of $k$ for which each of the following system of equations have infinitely many solution:
$
  8x + 5y = 9 \\
  kx + 10y = 18 \\
 $

Answer 1

Hint:– We will start solving this question by converting the given equations in matrix form and by using the given condition, i.e., equations have infinitely many solution, we can apply a formula, i.e., $\left( {adj.A} \right)B = 0$, from an application of matrix and determinants, i.e., linear system of equations, and obtain the required result.

Complete Step-by-Step solution:
The given system of equations is
\[
  8x + 5y = 9 \\
  kx + 10y = 18 \\
\]
Which can be written as
\[
  8x + 5y - 9 = 0 \\
  kx + 10y - 18 = 0 \\
\]
This system of equations is of the form
\[
  {a_1}x + {b_1}y + {c_1} = 0 \\
  {a_2}x + {b_2}y + {c_2} = 0 \\
\]
Where, \[{a_1} = 8,{b_1} = 5,{c_1} = - 9\] and \[{a_2} = k,{b_2} = 10,{c_2} = - 18\]
We know that for infinitely many solutions, we must have
\[
   \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} \\
   \Rightarrow \dfrac{8}{k} = \dfrac{5}{{10}} = \dfrac{{ - 9}}{{ - 18}} \\
\]
Now taking first and second parts of the above equation, we get
\[
   \Rightarrow \dfrac{8}{k} = \dfrac{5}{{10}} \\
   \Rightarrow \dfrac{8}{k} = \dfrac{1}{2} \\
  \therefore k = 8 \times 2 = 16 \\
\]
Thus, the required value of \[k\] is 16.

Note: – A matrix is an ordered rectangular array of numbers or functions. The system of equations is said to be consistent if it has one or more than one solutions. These types of questions become complex when we solve them using matrix formulas, to solve it perfectly, without making a mistake, all the basic formulas must be remembered.