Question

Find the value(s) of $p$ in the following pair of equations: $-3x+5y=7$ and $2px-3y=1$, if the lines represented by these equations are intersecting at a unique point.

Answer 1

Step 1: Compute the ratios of coefficients.

Given that: pair of linear equations is,

$$\begin{gathered} -3x+5y=7 \\ 2px-3y=1 \end{gathered}$$

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

$$\begin{gathered} a_1=-3,b_1=5,c_1=-7 \\ {a}_2=2p,b_2=-3,c_2=-1 \end{gathered}$$

$$\begin{gathered} \frac{a_1}{a_2}=\frac{-3}{2p} \\ \frac{b_1}{b_2}=\frac{-5}{3} \\ \frac{c_1}{c_2}=\frac{7}{-1} \end{gathered}$$

Step 2: Compute the required value.

Now, Since the lines are intersecting at a unique point i.e., it has a unique solution.

The condition that satisfies the unique solution is:

$$\begin{gathered} \frac{a_1}{a_2}\ne \frac{b_1}{b_2} \\ \Rightarrow \frac{-3}{2p}\ne \frac{-5}{3} \\ \Rightarrow p\ne \frac{9}{10} \end{gathered}$$

Hence, the lines represented by these equations are intersecting at a unique point for all real values of $p$ except $\frac{9}{10}$.