Whakaoti i te {l}{xsqrt{2}+ysqrt{3}=5}{xsqrt{3}-ysqrt{2}=0}

$$\left\{ \begin{array} { l } { x \sqrt { 2 } + y \sqrt { 3 } = 5 } \\ { x \sqrt { 3 } - y \sqrt { 2 } = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\sqrt{2}\approx 1.414213562$$

$$y=\sqrt{3}\approx 1.732050808$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakaraupapatia anō ngā kīanga tau.

$$\sqrt{3}x-\sqrt{2}y=0$$

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$\sqrt{2}x+\sqrt{3}y=5,\sqrt{3}x+\left(-\sqrt{2}\right)y=0$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$\sqrt{2}x+\sqrt{3}y=5$$

Me tango \sqrt{3}y mai i ngā taha e rua o te whārite.

$$\sqrt{2}x=\left(-\sqrt{3}\right)y+5$$

Whakawehea ngā taha e rua ki te \sqrt{2}.

$$x=\frac{\sqrt{2}}{2}\left(\left(-\sqrt{3}\right)y+5\right)$$

Whakareatia \frac{\sqrt{2}}{2} ki te -\sqrt{3}y+5.

$$x=\left(-\frac{\sqrt{6}}{2}\right)y+\frac{5\sqrt{2}}{2}$$

Whakakapia te \frac{-\sqrt{6}y+5\sqrt{2}}{2} mō te x ki tērā atu whārite, \sqrt{3}x+\left(-\sqrt{2}\right)y=0.

$$\sqrt{3}\left(\left(-\frac{\sqrt{6}}{2}\right)y+\frac{5\sqrt{2}}{2}\right)+\left(-\sqrt{2}\right)y=0$$

Whakareatia \sqrt{3} ki te \frac{-\sqrt{6}y+5\sqrt{2}}{2}.

$$\left(-\frac{3\sqrt{2}}{2}\right)y+\frac{5\sqrt{6}}{2}+\left(-\sqrt{2}\right)y=0$$

Tāpiri -\frac{3\sqrt{2}y}{2} ki te -\sqrt{2}y.

$$\left(-\frac{5\sqrt{2}}{2}\right)y+\frac{5\sqrt{6}}{2}=0$$

Me tango \frac{5\sqrt{6}}{2} mai i ngā taha e rua o te whārite.

$$\left(-\frac{5\sqrt{2}}{2}\right)y=-\frac{5\sqrt{6}}{2}$$

Whakawehea ngā taha e rua ki te -\frac{5\sqrt{2}}{2}.

$$y=\sqrt{3}$$

Whakaurua te \sqrt{3} mō y ki x=\left(-\frac{\sqrt{6}}{2}\right)y+\frac{5\sqrt{2}}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-\frac{\sqrt{6}}{2}\right)\sqrt{3}+\frac{5\sqrt{2}}{2}$$

Whakareatia -\frac{\sqrt{6}}{2} ki te \sqrt{3}.

$$x=\frac{-3\sqrt{2}+5\sqrt{2}}{2}$$

Tāpiri \frac{5\sqrt{2}}{2} ki te -\frac{3\sqrt{2}}{2}.

$$x=\sqrt{2}$$

Kua oti te pūnaha te whakatau.

$$x=\sqrt{2},y=\sqrt{3}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Whakaraupapatia anō ngā kīanga tau.

$$\sqrt{3}x-\sqrt{2}y=0$$

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$\sqrt{2}x+\sqrt{3}y=5,\sqrt{3}x+\left(-\sqrt{2}\right)y=0$$

Kia ōrite ai a \sqrt{2}x me \sqrt{3}x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \sqrt{3} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \sqrt{2}.

$$\sqrt{3}\sqrt{2}x+\sqrt{3}\sqrt{3}y=\sqrt{3}\times 5,\sqrt{2}\sqrt{3}x+\sqrt{2}\left(-\sqrt{2}\right)y=0$$

Whakarūnātia.

$$\sqrt{6}x+3y=5\sqrt{3},\sqrt{6}x-2y=0$$

Me tango \sqrt{6}x-2y=0 mai i \sqrt{6}x+3y=5\sqrt{3} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\sqrt{6}x+\left(-\sqrt{6}\right)x+3y+2y=5\sqrt{3}$$

Tāpiri \sqrt{6}x ki te -\sqrt{6}x. Ka whakakore atu ngā kupu \sqrt{6}x me -\sqrt{6}x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y+2y=5\sqrt{3}$$

Tāpiri 3y ki te 2y.

$$5y=5\sqrt{3}$$

Whakawehea ngā taha e rua ki te 5.

$$y=\sqrt{3}$$

Whakaurua te \sqrt{3} mō y ki \sqrt{3}x+\left(-\sqrt{2}\right)y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\sqrt{3}x+\left(-\sqrt{2}\right)\sqrt{3}=0$$

Whakareatia -\sqrt{2} ki te \sqrt{3}.

$$\sqrt{3}x-\sqrt{6}=0$$

Me tāpiri \sqrt{6} ki ngā taha e rua o te whārite.

$$\sqrt{3}x=\sqrt{6}$$

Whakawehea ngā taha e rua ki te \sqrt{3}.

$$x=\sqrt{2}$$

Kua oti te pūnaha te whakatau.

$$x=\sqrt{2},y=\sqrt{3}$$