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The system of linear equations 5x+ my=10 and 4x + ny=8 have infinitely many solutions,where m and n are positive Integers. Then the minimum possible value of (m n) is equal to.... 1)9
2)5
3)6
4)10


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The system of linear equations 5x+ my=10 and 4x + ny=8 have infinitel...
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Method:


  • Given system of linear equations are 5x - my = 10 and 4x - ny = 8

  • Since the given system of linear equations has infinitely many solutions, the lines are coincident.

  • So, the slopes of the lines must be equal.

  • Therefore,  5/m = 4/n = k (let)

  • => m/5 = n/4 = 1/k

  • => m = 5/k and n = 4/k

  • Also, 5x - my = 10 => y = (5x - 10)/m

  • Substituting this value of y in 4x - ny = 8, we get 4x - 4(5x - 10)/m = 8

  • => 4x - 20x/m + 10/m = 8

  • => 4mx - 20x + 10 = 8m

  • => x = (8m - 10)/(4m - 20) = 2/(m - 5)

  • Since x and y are positive integers,

  • 2/(m - 5) and (5x - 10)/m must be integers.

  • => m - 5 = 2 or 1

  • => m = 7 or 6

  • => n = 4/7 or 4/6

  • But m and n are positive integers. Therefore, m = 6 and n = 2

  • Therefore, the minimum possible value of mn is 6 x 2 = 12.



Answer: 12
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The system of linear equations 5x+ my=10 and 4x + ny=8 have infinitel...
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