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 = 5xyand 3x - 2y = xy. |
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Answer» Solution :(i) Given equations are8x - 3y = 5xy and6x - 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 = 5implies-3b = 5 - (124)/(11)` implies`-3b = - (69)/(11) implies b = (23)/(11)` Now,`(1)/(y) = a implies(1)/(y) = (31)/(22)impliesy = (22)/(31)` and `(1)/(x) = b implies(1)/(x) = (23)/(11)impliesx = (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 = 13impliesv = 1 Putting v = 1 in equation (5), we get 2(1) + 3u = 5implies3u = 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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