find the value of alpha and beta for which these equations have infinite solutions. 2x+3y=7 and 2alphax+(alpha+beta)y=5xy

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Solution

$The given equations are : 2 x + 3 y = 7 2 αx + (α + β) y = 5 Comparing above equations with a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0, we get a_{1} = 2; b_{1} = 3; c_{1} = - 7 a_{2} = 2 α, b_{2} = α + β, c_{2} = - 5 For infinite solution \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \frac{2}{2 α} = \frac{3}{(α + β)} = \frac{7}{5} from first and third, α = \frac{5}{7} from second and third, α + β = \frac{15}{7} β = \frac{15}{7} - \frac{5}{7} = \frac{10}{7}$

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