Find the value(s) of \( p \) and \( q \) for the following pair of equations:
\( 2 x+3 y=7 \) and \( 2 p x+p y=28-q y \),
if the pair of equations have infinitely many solutions.
Given:
The given system of equations is:
\( 2 x+3 y=7 \) and \( 2 p x+p y=28-q y \),
To do:
We have to find the values of \( p \) and \( q \) for which the given system of equations has infinitely many solutions.
Solution:
The given system of equations can be written as:
$2x + 3y -7=0$
$2px +(p+q)y -28=0$
The standard form of system of equations of two variables is $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y-c_{2}=0$.
The condition for which the above system of equations has infinitely many solutions is
$\frac{a_{1}}{a_{2}} =\frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
Comparing the given system of equations with the standard form of equations, we have,
$a_1=2, b_1=3, c_1=-7$ and $a_2=2p, b_2=p+q, c_2=-28$
Therefore,
$\frac{2}{2p}=\frac{3}{p+q}=\frac{-7}{-28}$
$\frac{1}{p}=\frac{1}{4}$
$p=4$
$\frac{3}{p+q}=\frac{1}{4}$
$4\times 3=1(p+q)$
$p+q=12$
$4+q=12$
$q=12-4=8$
The values of $p$ and $q$ for which the given system of equations has infinitely many solutions are $4$ and $8$ respectively.
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