Question

If the lines represented by the equations 3x – y – 5 = 0 and 6x – 2y – p = 0 are parallel, then p is equal to ________.

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 no solution, i.e., they are parallel, then $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$.

Since, the system of equations 3x – y – 5 = 0 and 6x – 2y – p = 0 are parallel
Therefore,
$$\begin{gathered} \frac{3}{6}=\frac{-1}{-2}\ne \frac{-5}{-p} \\ \Rightarrow \frac{1}{2}=\frac{1}{2}\ne \frac{5}{p} \\ \Rightarrow \frac{1}{2}\ne \frac{5}{p} \\ \Rightarrow p\ne 10 \end{gathered}$$

Hence, if the lines represented by the equations 3x – y – 5 = 0 and 6x – 2y – p = 0 are parallel, then p is equal to all real values except 10.