Question

The value of c for which the pair of equations \[cx-y=2\] and \[6x-2y=3\] will have infinitely many solutions is
A. \[3\]
B. \[-3\]
C. \[-12\]
D. no value

Answer 1

Hint:In this question, we need to find the value of \[c\]. Also given a pair of equations \[cx-y=2\] and \[6x-2y= 3\] which have infinitely many solutions. First, Let us rewrite both the equations such that their right hand sides are equal to zero. Then let us know the concept of the consistent equations. We can use this concept to reach the solution of the given problem.

Complete step by step answer:
Given two equations \[cx-y=2\] and \[6x-2y= 3\] which have infinitely many solutions.Let us rewrite both the equations such that their right hand sides are equal to zero.\[cx-y=2\] can be rewritten as \[cx-y-2 = 0\] and also \[6x-2y= 3\] as \[6x-2y-3 = 0\]. Now, let us know the concept of the consistent equations.Let us consider two equations \[a_{1}x + b_{1}y + c_{1} = 0\] and \[a_{2}x + b_{2}y + c_{2} = 0\] .We can tell that the lines will have infinitely many solutions when the lines are coincident.

If the two lines are coincident,
That is \[\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} = \dfrac{c_{1}}{c_{2}}\]
Also if the lines are not coincident,
That is \[\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} \neq \dfrac{c_{1}}{c_{2}}\]
Here are two equations \[cx-y-2 = 0\] and \[6x-2y-3=0\] which have infinitely many solutions.
Here \[a_{1} = c\] , \[b_{1} = - 1\] , \[c_{1} = - 2\] and also \[a_{2} = 6\] , \[b_{2} = - 2\] , \[c_{2} = - 3\]

Now by using the condition of consistent equations.
We get,
\[\Rightarrow \dfrac{a_{1}}{a_{2}} = \dfrac{c}{6}\] ,\[\dfrac{b_{1}}{b_{2}} = - \dfrac{1}{- 2}\] and \[\dfrac{c_{1}}{c_{2}} = - \dfrac{2}{- 3} = \dfrac{2}{3}\]
On observing,
We can tell that \[\dfrac{b_{1}}{b_{2}} \neq \dfrac{c_{1}}{c_{2}}\]
That means that the lines are not coincident. So we can’t find the value of \[c\].

Hence, the correct answer is option D.

Note:In order to solve these types of questions, we should have a strong grip over the consistent equations. We need to know that for no value of \[c\] the pair of equations will have infinitely many solutions. Mathematically , a linear or nonlinear system of equations is known as the consistent equations if there is at least one set of values for the unknowns that satisfies all of the equations. Similarly, a linear or nonlinear system of equations is known as the inconsistent equations if there is no set of values for the unknowns that satisfies all of the equations.