Question

Find the value of “k” for which the given set of linear equations has infinite solutions. The given equations are:
\[
  x + (k + 1)y = 5 \\
  (k + 1)x + 9y = 8k - 1 \;
 \]

Answer 1

Hint: To get an infinite solution the given equation should be equal, because when two system of equations are equal then they are defined for each and every value possible which is infinite in case for linear equations, so for getting the answer you have to equate the equations.

Complete step-by-step answer:
The given set of questions are:
\[
  x + (k + 1)y = 5 \\
  (k + 1)x + 9y = 8k - 1 \;
 \]
We know that for getting the value of “k” for which both the equations have infinite solution we have to equate both the equations, on equating the equations we have to make some rearrangements, doing that we get:
\[
   \Rightarrow x + (k + 1)y - 5 = 0 \\
   \Rightarrow (k + 1)x + 9y - 8k + 1 = 0 \;
 \]
On solving these equation now we get:
\[
   \Rightarrow x + (k + 1)y - 5 = (k + 1)x + 9y - 8k + 1 \\
   \Rightarrow x + ky + y - 5 = kx + x + 9y - 8k + 1 \\
   \Rightarrow ky - kx - 8k = x + 9y + 1 - x - y + 5 \\
   \Rightarrow k(y - x - 8) = 8y + 6 \\
   \Rightarrow k = \dfrac{{8y + 6}}{{y - x - 8}} \;
 \]
So this is our required value of “k” for which both the set of linear equations have infinite solutions.
So, the correct answer is “$ k = \dfrac{{8y + 6}}{{y - x - 8}}$”.

Note: Once you get the value of “k” put the value in the main equation, result will be left hand side is equal to right hand side, or just put any assumed value of the variable and check in the equation you will see that the equations are satisfying the assumed value.
In this kind of question where we have to solve for a single value using two or more equation the best way is to equate the equations, or you can get the value of one variable from one equation and then put it in next equation you will see that the solution obtained will be same in both the case.