Question

Find the value(s) of $p$ and $q$ in the following pair of equations: $2x+3y=7$ and $2px+3y=28-qy$ if the pair of equations have infinitely many solutions.

Answer 1

Step 1: Compute ratios of coefficients.

Given pair of linear equations is,

$$\begin{gathered} 2x+3y=0 \\ 2px+\left(p+q\right)y-28=0 \end{gathered}$$

On comparing with $ax+by+c=0$, we obtain

$$\begin{gathered} a_1=2,b_1=3,c_1=-7 \\ {a}_2=2p,b_2=p+q,c_2=-28 \end{gathered}$$

Also ratios of coefficients,

$$\begin{gathered} \frac{a_1}{a_2}=\frac{2}{2p} \\ \frac{b_1}{b_2}=\frac{3}{p+q} \\ \frac{c_1}{c_2}=\frac{1}{4} \end{gathered}$$

Step 2: Compute the required values:

Since the pair of equations has infinitely many solutions i.e., both lines are coincident.

$$\begin{gathered} \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \\ \Rightarrow \frac{1}{p}=\frac{3}{p+q}=\frac{1}{4} \end{gathered}$$

On considering the first and third parts, we obtain

$p=4$

On considering the last two parts, we obtain

$$\begin{gathered} \frac{3}{4+q}=\frac{1}{4} \\ \Rightarrow 4+q=12 \\ \Rightarrow q=8 \end{gathered}$$

Hence, the value of $p$ is $4$ and $q$ is $8$.