Find the values of $ a $ and $ b $ for which the following system of equations has infinitely many solutions:
$2x-(2a+5) y=5$
$(2b+1)x-9y=15$

Given: 

The given system of equations is:

$2x-(2a+5) y=5$
$(2b+1)x-9y=15$

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:

$2x-(2a+5)y-5=0$

$(2b+1)x-9y-15=0$

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

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

$a_1=2, b_1=-(2a+5), c_1=-5$ and $a_2=(2b+1), b_2=-9, c_2=-15$

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}} \ $

$\frac{2}{2b+1}=\frac{-(2a+5)}{-9}=\frac{-5}{-15}$

$\frac{2}{2b+1}=\frac{2a+5}{9}=\frac{1}{3}$

$\frac{2}{2b+1}=\frac{1}{3}$ and $\frac{2a+5}{9}=\frac{1}{3}$

$2\times3=1\times(2b+1)$ and $3\times(2a+5)=1\times9$

$6=2b+1$ and $6a+15=9$

$2b=6-1$ and $6a=9-15$

$2b=5$ and $6a=-6$

$b=\frac{5}{2}=3$ and $a=\frac{-6}{6}$

$b=\frac{5}{2}$ and $a=-1$

The values of $a$ and $b$ for which the given system of equations has infinitely many solutions is $-1$ and $\frac{5}{2}$ respectively.  

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Updated on: 10-Oct-2022

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