Tìm x, y biết: 13x^2 + 9y^2 - 12xy - 26x - 6y + 26 = 0

Thanh Truyen Le

$13x^2 + 9y^2 - 12xy - 26x - 6y + 26 = 0$

$y = z + \frac{2}{3}x$

Thay vào:

$13x^2 + 9\left(z + \frac{2}{3}x\right)^2 - 12x\left(z + \frac{2}{3}x\right) - 26x - 6\left(z + \frac{2}{3}x\right) + 26 = 0$

$13x^2 + 9\left(z^2 + \frac{4}{3}xz + \frac{4}{9}x^2\right) - 12x z - 8x^2 - 26x - 6z - 4x + 26 = 0$

$9x^2 + 9z^2 - 30x - 6z + 26 = 0$

$x^2 + z^2 - \frac{10}{3}x - \frac{2}{3}z + \frac{26}{9} = 0$

$\left(x - \frac{5}{3}\right)^2 + \left(z - \frac{1}{3}\right)^2 = 0$

Suy ra:

$x = \frac{5}{3}, \quad z = \frac{1}{3}$

Thay lại $z = y - \frac{2}{3}x$:

$y = \frac{1}{3} + \frac{2}{3} \cdot \frac{5}{3} = \frac{13}{9}$

$\boxed{x = \frac{5}{3}, \quad y = \frac{13}{9}}$