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Find the values of $p$ and $q$ if the pair of equations have infinitely many solutions.$2x+3y=7$ and $2px+py=28-qy$.
Given: The pair of equations have infinitely many solutions.$2x+3y=7$ and $2px+py=28-qy$.
To do: To find the values of $p$ and $q$.
Solution:
$2x+3y=7$ ------ $( 1)$
$2px+py=28-qy$
$\Rightarrow 2px+py+qy=28$
$\Rightarrow 2px+( p+q) y=28$ ------- $( 2)$
As given, equations have infinite Solutions.
$\Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
$\Rightarrow \frac{2}{2p}=\frac{3}{( p+q)}=\frac{7}{28}$ ------ $( 3)$
From 1st and 2nd part of the equation $( 3)$
$\frac{2}{2p}=\frac{7}{28}$
$\Rightarrow 14p=56$
$\Rightarrow p=4$
Now,
From 2nd and 3rd part of equation $( 3)$
$\Rightarrow \frac{3}{( p+q)}=\frac{7}{28}$
$\Rightarrow 3\times 28=7 ( p+q)$
$\Rightarrow 3\times 4=( p+q)$
$\Rightarrow 12=4+q$
$\Rightarrow q=12-4$
$\Rightarrow q=8$
Thus, $p=4$ and $q=8$.
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