System of equations

Algebra Level pending

If the values for m m for which the system of equations 3 x + m y = m 3x+my=m and 2 x − 5 y = 20 2x-5y=20 have solutions satisfying the conditions x > 0 , y > 0 x>0,\quad y>0 can be written as

m ∈ ( − ∞ , − a b ) ∪ ( c , ∞ ) m\in \left( -\infty ,\frac { -a }{ b } \right) \cup \left( c,\infty \right) , where a and b are positive coprime integers.

then find the value of a + b + c a+b+c


The answer is 47.

Krishna Ramesh by Krishna Ramesh , 22, India

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1 solution

Ronak Agarwal
Aug 12, 2014

Solve the linear equations to get y = m − 30 m + 15 2 , x = 25 m m + 15 2 y=\frac { m-30 }{ m+\frac { 15 }{ 2 } } ,x=\frac { 25m }{ m+\frac { 15 }{ 2 } }

Now x > 0 , y > 0 x>0,y>0 , applying wavy curve method we have :

m ϵ ( − ∞ , − 15 2 ) ∪ ( 30 , ∞ ) m\epsilon (-\infty ,\frac { -15 }{ 2 } )\cup (30,\infty )

Hence a = 15 , b = 2 , c = 30 a=15,b=2,c=30

⇒ a + b + c = 47 \Rightarrow \boxed{a+b+c=47}

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