Question

Find the value(s) of $p$ in the following pair of equations: $3x-y-5=0$ and $6x-2y-p=0$, if the lines represented by these equations are parallel.

Answer 1

Step 1: Compute the ratios of coefficients.

Given: pair of linear equations is

$$\begin{gathered} 3x-y-5=0 \\ 6x-2y-p=0 \end{gathered}$$

On comparing the given equations with $ax+by+c=0$ , we get

$$\begin{gathered} a_1=3,b_1=-1,c_1=-5 \\ {a}_2=6,b_2=-2,c_2=-p \end{gathered}$$

$$\begin{gathered} \frac{a_1}{a_2}=\frac{3}{6} \\ \frac{b_1}{b_2}=\frac{1}{2} \\ \frac{c_1}{c_2}=\frac{5}{p} \end{gathered}$$

Step 2: Compute the required value.

Since the lines represented by these equations are parallel.

Then the condition is:

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

On considering the last two parts, we get

$$\begin{gathered} \frac{1}{2}\ne \frac{5}{p} \\ \Rightarrow p\ne 10 \end{gathered}$$.

Hence, the given pair of linear equations are parallel for all real values of $p$ except $10$.