Question

There is (are) ________ value(s) of c for which the pair of equations cx – y = 2 and 6x – 2y = 3 have infinitely many solutions.

Answer 1

If the system of equations $a_{1}x+b_{1}y+c_1=0\mathrm{and}{a}_{2}x+b_{2}y+c_2=0$ has infinitely many solutions, then $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.

Since, the system of equations cx – y = 2 and 6x – 2y = 3 has infinitely many solutions
Therefore,
$$\begin{gathered} \frac{c}{6}=\frac{-1}{-2}=\frac{-2}{-3} \\ \Rightarrow \frac{c}{6}=\frac{1}{2}=\frac{2}{3} \\ \mathrm{But},\frac{1}{2}\ne \frac{2}{3} \end{gathered}$$

Therefore, for no value of c the pair of equations cx – y = 2 and 6x – 2y = 3 have infinitely many solutions.

Hence, there is (are) no value(s) of c for which the pair of equations cx – y = 2 and 6x – 2y = 3 have infinitely many solutions.