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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 $\frac{9}{10}$.