Matematik Tambahan SPM 2019, Kertas 2 (Soalan 1)

Soalan 1:
Selesaikan persamaan serentak x + 2y = 1 dan

\frac{3}{x} - \frac{2}{y} = 5

.
Beri jawapan anda betul kepada tiga tempat perpuluhan. [5 markah]

Penyelesaian:

\begin{aligned} x + 2 y = 1 \\ x = 1 - 2 y \dots \dots \dots \dots (1) \\ \frac{3}{x} - \frac{2}{y} = 5 \\ \times x y, 3 y - 2 x = 5 x y \dots \dots \dots \\ Ganti (1) dalam (2): \\ 3 y - 2 (1 - 2 y) = 5 y (1 - 2 y) \\ 3 y - 2 + 4 y = 5 y - 10 y^{2} \\ 10 y^{2} + 2 y - 2 = 0 \\ 5 y^{2} + y - 1 = 0 \\ a = 5, b = 1, c = - 1 \\ y = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \\ = \frac{- 1 \pm \sqrt{1 - 4 (5) (- 1)}}{10} \end{aligned}

\begin{aligned} \begin{matrix} y = \frac{- 1 + \sqrt{21}}{10} \\ = 0.3583 \\ = 0.358 (3 tp) \end{matrix} \\ atau \\ y = \frac{- 1 - \sqrt{21}}{10} \\ \begin{matrix} = - 0.5583 \\ = - 0.558 (3 tp) \\ Dari (1) \\ Apabila y = 0.358 \\ x = 1 - 2 (0.3583) \\ = 0.283 \\ Apabila y = - 0.558 \\ x = 1 - 2 (- 0.5583) \\ = 2.117 \end{matrix} \end{aligned}