InterviewSolution
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InterviewSolution
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(i) Solve 8x - 3y = 5xy and 6x - 5y = - 2xy. How many solutions, this system of equations has? (ii) Solve for x and y, by reducing the following equations in a pair of linear equations : 2x + 3y = 5xy and 3x - 2y = xy. |
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Answer» (i) Given equations are 8x - 3y = 5xy and 6x - 5y = -2xy The given equations are not linear in the variable x and y. These can be reduced into linear equations. If we put x = 0 in either of the equations, we get y = 0. Hence, x = 0 and y = 0 is a solution of these equations. To find other solution, dividing each of th given equations by xy. So, `(8)/(y) - (3)/(x) = 5` .... (1) and `(6)/(y) - (5)/(x) = -2` ....(2) Let `(1)/(y)` = a and `(1)/(x)` = b then from equations (1) and (2), we get `{:(8a - 3b = 5),(6a - 5b =-2):}}"linear equations "{:(....(3)),( ....(4)):}` Multiplying equation (3) by 5 and (4) by 3, we get 40a - 15b = 25 ....(5) 18a - 15b = - 6 ....(6) Subtracting equation (6) from (5), we get 22a = 31 implies `a = (31)/(22)` Putting `a = (31)/(22)` in equation (3), we get `8 xx (31)/(22) - 3b = 5` `(124)/(11) - 3b = 5 implies -3b = 5 - (124)/(11)` implies `-3b = - (69)/(11) implies b = (23)/(11)` Now, `(1)/(y) = a implies (1)/(y) = (31)/(22) implies y = (22)/(31)` and `(1)/(x) = b implies (1)/(x) = (23)/(11) implies x = (11)/(23)` Hence, the solution of the given two equations are `{:(x = 0),(y = 0):}}` and `{:(x = (11)/(23)),(y = (22)/(31)):}}`. So, the given system of equations has two solutions. (ii) We have, 2x + 3y = 5xy .... (1) 3x - 2y = xy .... (2) Dividing equations (1) and (2) by xy, we get `(2x)/(xy) + (3y)/(xy) = (5xy)/(xy) implies (2)/(y) + (3)/(x) = 5 " "...(3)` and `(3x)/(xy) - (2y)/(xy) = (xy)/(xy) implies (3)/(y) - (2)/(x) = 1 " "....(4)` Let `(1)/(x) = u` and `(1)/(y) = v`. `therefore` Equations (3) and (4) become, 2v + 3u = 5 ....(5) 3v - 2u = 1 ....(6) Now this is a pair of linear equations,multiplying equation (5) by 2 and equation (6) by 3, we get 4v + 6u = 10 ....(7) 9v - 6u = 3 ....(8) On adding equations (7) and (8), we get 13v = 13 implies v = 1 Putting v = 1 in equation (5), we get 2(1) + 3u = 5 implies 3u = 3 implies u = 1 Now, u = 1 implies `(1)/(x) = 1` implies x = 1 and v = 1 implies `(1)/(y) = 1` implies y = 1 Hence, required solution is `{:(x = 1),(y = 1):}}`. |
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