Question

for what value of A and B will the following pair of linear equations have infinitely many solutions
x+2y=1 , (a-b)x+(a+b)y=a+b-2 with explanation

Answer 1

hope that answers your question

Answer 2

Answer:  The required values are  a = 3  and  b = 1.

Step-by-step explanation:  We are given to find the values of a and b for which the following pair of linear equations will have infinitely many solutions :

$x+2y=1,\\\\(a-b)x+(a+b)y=a+b-2.$

We know that

for a system of linear equations to have infinitely many solutions, the coefficients of the unknown variables (x and y) and the constant terms must be in proportion.

So, for the given system, we must have

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

We have from above that

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

and

$\dfrac{2}{a+b}=\dfrac{1}{a+b-2}\\\\\Rightarrow 2(a+b-2)=a+b\\\\\Rightarrow 2a+2b-4=a+b\\\\\Rightarrow 2a-a+2b-b=4\\\\\Rightarrow a+b=4\\\\\Rightarrow 4b=4~~~~~~~~~~~~~~~~~~~~~~~[\textup{Uisng equation (i)}]\\\\\Rightarrow b=\dfrac{4}{4}\\\\\Rightarrow b=1$

Again, substituting the value of b in equation (i), we get

$a=3\times1=3.$

Thus, the required values are  a = 3  and  b = 1.