Question

find the value of a and b for which each of the following system of linear equations has a infinite number of solutions: 1) 2x-3y=7; (a+b)x-(a+b-3)y=4a​

Answer 1

The two equations given are, 2 x-3 y=7 and (a+b) x-(a+b-3) y=4 a+b

Comparing the two to get infinite value is

$\frac{2}{a+b}=\frac{-3}{a+b-3}=\frac{7}{4 a+b}$

Solve the two equations one by one first separate

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

a+b=−6

Now solving the second half we get

$\begin{gathered}\begin{array}{l}{\frac{-3}{a+b-3}=\frac{7}{4 a+b}} \\ {-3(4 a+b)=7(a+b-3)} \\ {a=5 b}\end{array}\end{gathered}$

Solving the two halves we get the value of a and b which is

$\begin{gathered}\begin{array}{l}{a+b=-6, a=5 b} \\ {5 b+b=-6} \\ {b=-1}\end{array}\end{gathered}$

Putting the value of b in a+b=-6a+b=−6 , we get

$\begin{gathered}\begin{array}{l}{a-1=-6} \\ {a=-5}\end{array}\end{gathered}$

Therefore, the value of “a” and “b” is equal to -5 and -1 respectively.