Bab 4 Persamaan Serentak

Posted on October 10, 2016 by user

4.2 SPM Praktis, Persamaan Serentak, (Soalan panjang)

Soalan 5:

Selesaikan persamaan serentak.

5y – 6x= 2

\frac{4 y}{x} - \frac{3 x}{y} = 4.

Penyelesaian:

5y – 6x= 2 ----- (1)

\begin{array}{l} \frac{4 y}{x} - \frac{3 x}{y} = 4 ------- (2) \\ Daripada (1), \\ y = \frac{2 + 6 x}{5} ------- (3) \end{array}

Gantikan (3) dalam (2),

\begin{array}{l} \frac{4 (\frac{2 + 6 x}{5})}{x} - \frac{3 x}{(\frac{2 + 6 x}{5})} = 4 \\ \frac{8 + 24 x}{5 x} - \frac{15 x}{2 + 6 x} = 4 \\ \frac{(8 + 24 x) (2 + 6 x) - (15 x) (5 x)}{5 x (2 + 6 x)} = 4 \end{array}

16 + 48x + 48x + 144x² – 75x² = 20x (2 + 6x)

69x² + 96x + 16 = 40x + 120x²

51x² – 56x – 16 = 0

(3x – 4)(17x + 4) = 0

3x – 4 = 0 atau 17x + 4 = 0

x = \frac{4}{3} atau x = - \frac{4}{17}

\begin{array}{l} Gantikan x = \frac{4}{3} dalam persamaan (3), \\ y = \frac{2 + 6 (\frac{4}{3})}{5} = 2 \end{array}

\begin{array}{l} Gantikan x = - \frac{4}{17} dalam persamaan (3), \\ y = \frac{2 + 6 (- \frac{4}{17})}{5} = \frac{2}{17} \end{array}

Maka penyelesaian ialah (\frac{4}{3}, 2) dan (- \frac{4}{17}, \frac{2}{17}) .

Soalan 6:
Selesaikan persamaan serentak.

\begin{array}{l} x + 2 y = 1 \\ 2 x^{2} + y^{2} + x y = 5 \end{array}

Beri jawapan anda betul kepada tiga tempat perpuluhan.

Penyelesaian:

\begin{array}{l} x + 2 y = 1.......... (1) \\ 2 x^{2} + y^{2} + x y = 5.......... (2) \\ x = 1 - 2 y .......... (3) \\ Gantikan (3) ke dalam (2), \\ 2 {(1 - 2 y)}^{2} + y^{2} + (1 - 2 y) y = 5 \\ 2 (1 - 4 y + 4 y^{2}) + y^{2} + y - 2 y^{2} = 5 \\ 2 - 8 y + 8 y^{2} - y^{2} + y - 5 = 0 \\ 7 y^{2} - 7 y - 3 = 0 \\ a = 7, b = - 7, c = - 3 \\ x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \\ y = \frac{- (- 7) \pm \sqrt{{(- 7)}^{2} - 4 (7) (- 3)}}{2 (7)} \\ y = \frac{7 \pm \sqrt{133}}{14} \\ y = 1.324 atau - 0.324 \\ Dari x = 1 - 2 y \\ Apabila y = 1.324, \\ x = 1 - 2 (1.324) = 1.648 \\ Apabila y = - 0.324, \\ x = 1 - 2 (- 0.324) = 1.648 \\ ∴ Penyelesaian ialah (1.648, 1.324) dan (1.648, - 0.324) . \end{array}