Question

for which values of A and B will the following pair of linear equation have infinitely many solutions?
X + 2 Y =1
(a-b)x + (a+b)y= a+b-2

Answer 1

Dear Student,

$$\begin{gathered} Thegivensystemofequationsis- \\ x+2y=1 \\ x+2y-1=0....\left(1\right) \\ \left(a-b\right)x+\left(a+b\right)y=a+b-2 \\ \left(a-b\right)x+\left(a+b\right)y-\left(a+b-2\right)=0.....\left(2\right) \\ Forthesystemofequationtohaveinfinitelymanysolutions,\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \\ \frac{1}{\left(a-b\right)}=\frac{2}{\left(a+b\right)}=\frac{-1}{-\left(a+b-2\right)} \\ Fromfirsttwoterms,weget \\ \frac{1}{\left(a-b\right)}=\frac{2}{\left(a+b\right)} \\ a+b=2a-2b \\ a-3b=0....\left(3\right) \\ Fromfirstandthirdterm,weget \\ \frac{1}{\left(a-b\right)}=\frac{1}{\left(a+b-2\right)} \\ a+b-2=a-b \\ 2b=2 \\ \mathit{b}\mathbf{=}\mathbf{1} \\ substitutingvalueofbinequ.\left(3\right),weget \\ \mathit{a}\mathbf{=}\mathbf{3} \end{gathered}$$

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