Question

find the value of a and b for the following system of linear equations to have a infinite number of solutions
1) 3x-y=14 ; (a+b)x-2by=5a+2b+1

Answer 1

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Answer 2

Answer:  The required values are  a = 5  and  b = 1.

Step-by-step explanation:  We are given to find the values of a and b for which the following system of equations will have an infinite number of solutions :

$3x-y=14~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\(a+b)x-2by=5a+2b+1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

For the above system of equations to have an infinite number of solutions, the coefficients of x, y and the constant terms must be proportional.

That is,

$\dfrac{3}{a+b}=\dfrac{-1}{-2b}=\dfrac{14}{5a+2b+1}.$

We have

$\dfrac{3}{a+b}=\dfrac{-1}{-2b}\\\\\\\Rightarrow \dfrac{3}{a+b}=\dfrac{1}{2b}\\\\\Rightarrow 6b=a+b\\\\\Rightarrow a=6b-b\\\\\Rightarrow a=5b~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)$

and

$\dfrac{-1}{-2b}=\dfrac{14}{5a+2b+1}\\\\\\\Rightarrow \dfrac{1}{2b}=\dfrac{14}{5a+2b+1}\\\\\Rightarrow 5a+2b+1=28b\\\\\Rightarrow 5a=28b-2b-1\\\\\Rightarrow 5a=26b-1\\\\\Rightarrow a=\dfrac{26b-1}{5}~~~~~~~~~~~~~~~~~~~~~~~~~~(iv)$

Comparing equations (iii) and (iv), we get

$5b=\dfrac{26b-1}{5}\\\\\Rightarrow 25b=26b-1\\\\\Rightarrow 26b-25b=1\\\\\Rightarrow b=1.$

From equation (iii), we get

$a=5\times1=5.$

Thus, the required values are  a = 5  and  b = 1.