Find the values of $ a $ and $ b $ for which the following system of equations has infinitely many solutions:
$2x+3y=7$
$2ax+ay=28-by$

Given: 

The given system of equations is:

$2x+3y=7$
$2ax+ay=28-by$

To do: 

We have to determine the values of $a$ and $b$ so that the given system of equations has infinitely many solutions.

Solution:

The given system of equations can be written as:

$2x+3y-7=0$.....(i)
$2ax+ay-28+by=0$

$2ax+(a+b)y-28=0$.......(ii)

The standard form of system of equations of two variables is $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y-c_{2}=0$.

Comparing the given system of equations with the standard form of equations, we have,

$a_1=2, b_1=3, c_1=-7$ and $a_2=2a, b_2=a+b, c_2=-28$

The condition for which the given system of equations has infinitely many solutions is

$\frac{a_{1}}{a_{2}} \ =\frac{b_{1}}{b_{2}} =\frac{c_{1}}{c_{2}} \ $

$\frac{2}{2a}=\frac{3}{a+b}=\frac{-7}{-28}$

$\frac{1}{a}=\frac{1}{4}$ and $\frac{3}{a+b}=\frac{1}{4}$

$1\times(4)=1\times(a)$ and $3\times(4)=1\times(a+b)$

$a=4$ and $a+b=12$

Using $a=4$ in $a+b=12$, we get,

$4+b=12$

$b=12-4=8$

The values of $a$ and $b$ for which the given system of equations has infinitely many solutions is $4$ and $8$ respectively.

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Updated on: 10-Oct-2022

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