Increasing Exponential System of Equations

Algebra Level 5

{ a x + b y = 4 a x 2 + b y 2 = 2 a x 3 + b y 3 = − 3 ( 2 x − 1 ) ( 2 y − 1 ) = ? \begin{cases} ax+by & = 4 \\ ax^2+by^2 & = 2 \\ ax^3+by^3 & =-3 \\ (2x-1)(2y-1) & =~~? \end{cases}


The answer is 4.

Rohit Udaiwal by Rohit Udaiwal , 20, India

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1 solution

Rohit Udaiwal
Apr 17, 2016

We have { a x + b y = 4 … ( 1 ) a x 2 + b y 2 = 2 … ( 2 ) a x 3 + b y 3 = − 3 … ( 3 ) \begin{cases} ax+by & = 4 \ldots(1) \\ ax^2+by^2 & = 2\ldots(2) \\ ax^3+by^3 & =-3\ldots(3) \end{cases} Next note that ( a x 2 + b y 2 ) ( x + y ) − ( a x + b y ) x y = a x 3 + b y 3 ⟹ − 3 = 2 ( x + y ) − 4 x y [ Using 1 , 2 , 3. ] ⟹ 4 x y − 2 ( x + y ) + 1 = 3 + 1 ∴ ( 2 x − 1 ) ( 2 y − 1 ) = 4 \begin{aligned} (ax^2+by^2)(x+y)-(ax+by)xy= & ax^3+by^3 \\ \implies -3= & 2(x+y)-4xy \quad \quad [\text{Using}~ 1,2,3.] \\ \implies 4xy-2(x+y)+1= & 3+1 \\ \therefore (2x-1)(2y-1) = & \boxed{4} \end{aligned}

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https://brilliant.org/problems/an-easy-one-19/?ref_id=1255515
same question was posted by me .. sir can you please tell from where do you got this question ....

Deepansh Jindal - 4 years, 9 months ago

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Hello sir, this question appeared in the periodical of a certain coaching institute. :)

Rohit Udaiwal - 4 years, 9 months ago

I think the coaching institute is resonance

Deepansh Jindal - 4 years, 9 months ago

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