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 $q=8$.

Updated on: 10-Oct-2022

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