Isn't Familiar! Part III

Algebra Level 5

{ 4 x 3 − 3 x y = 9 x + ( x + 3 ) 3 3 x 2 − 2 y + 1 = 7 x − 2 x + 1 \left\{\begin{matrix} 4x^3-3xy=9x+\sqrt{(x+3)^3} & & \\ 3x^2-2y+1=7x-2\sqrt{x+1} & & \end{matrix}\right.

How many pairs of ( x , y ) (x,y) satisfy the system of equations above?


The answer is 3.

Son Nguyen by Son Nguyen , 20, Vietnam

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1 solution

Chew-Seong Cheong
Sep 19, 2018

{ 4 x 3 − 3 x y = 9 x + ( x + 3 ) 3 . . . ( 1 ) 3 x 3 − 2 y + 1 = 7 x − 2 x + 1 . . . ( 2 ) \begin{cases} 4x^3 - 3xy = 9x + \sqrt{(x+3)^3} & ...(1) \\ 3x^3 - 2y + 1 = 7x - 2\sqrt{x+1} & ...(2) \end{cases}

3 x ( 2 ) − 2 ( 1 ) : x 3 + 3 x = 21 x 2 − 6 x x + 1 − 18 x − 2 ( x + 3 ) 3 \begin{aligned} 3x(2)-2(1): \quad x^3 + 3x & = 21x^2 - 6x\sqrt{x+1} - 18 x - 2\sqrt{(x+3)^3} \end{aligned}

⟹ x 3 − 21 x 2 + 21 x + 6 x x + 1 + 2 ( x + 3 ) 3 = 0 \begin{aligned} \implies x^3-21x^2+21x + 6x\sqrt{x+1} + 2\sqrt{(x+3)^3} & = 0 \end{aligned}

The equation above has three roots, therefore there are 3 \boxed 3 pairs of ( x , y ) (x,y) satisfying the system of equations.

Solving the equation above, the three solution pairs are ( − 0.27995 , 2.44595 ) , ( 2.26951 , 2.09091 ) , ( 17.7537 , 415.483 ) (-0.27995, 2.44595), (2.26951, 2.09091), (17.7537, 415.483) .

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