Question

Find the value(s) of $p$ in the following pair of equations: $2x+3y-5=0$ and $px-6y-8=0$, if the pair of equations has a unique solution.

Answer 1

Step 1: Compute the ratio of coefficients.

Given pair of linear equations is:

$$\begin{gathered} 2x+3y-5=0 \\ px-6y-8=0 \end{gathered}$$

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

$$\begin{gathered} a_1=2,b_1=3,c_1=-5 \\ {a}_2=p,b_2=-6,c_2=-8 \end{gathered}$$

Now, the ratios of coefficients,

$$\begin{gathered} \frac{a_1}{a_2}=\frac{2}{p} \\ \frac{b_1}{b_2}=\frac{-3}{6}=\frac{-1}{2} \\ \frac{c_1}{c_2}=\frac{5}{8} \end{gathered}$$

Step 2 : Compute the required value:

Since the pair of linear equations has a unique solution.

$$\begin{gathered} \Rightarrow \frac{a_1}{a_2}\ne \frac{b_1}{b_2} \\ \Rightarrow \frac{2}{p}\ne \frac{-1}{2} \\ \Rightarrow p\ne -4 \end{gathered}$$

Hence, the pair of linear equations has a unique solution for all values of $p$ except $-4$.