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

Given:

Given pair of linear equations is:

\( -3 x+5 y=7 \) and \( 2 p x-3 y=1 \).

To do:

We have to find the value of $p$ if the given system of equations are intersecting at a unique point.

Solution:

Comparing the given pair of linear equations with the standard form of linear equations $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$, we get,

$a_1=-3, b_1=5$ and $c_1=-7$

$a_2=2p, b_2=-3$ and $c_2=-1$

A system of equations has a unique solution if it satisfies the following condition,

$\frac{a_1}{a_2}≠ \frac{b_1}{b_2}$

Here,

$\frac{a_1}{a_2}=\frac{-3}{2p}$

$\frac{b_1}{b_2}=\frac{5}{-3}=-\frac{5}{3}$

Therefore,

$\frac{a_1}{a_2}≠\frac{b_1}{b_2}$

$\frac{-3}{2p}≠\frac{-5}{3}$

$-3(3)≠-5\times2p$

$-9≠-10p$

$p≠\frac{9}{10}$

Therefore, the values of $p$ are all real values except $\frac{9}{10}$.

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Updated on: 10-Oct-2022

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