Whakaoti i te {l}{xsqrt{2}-ysqrt{5}=2sqrt{10}}{xsqrt{5}+ysqrt{2}=3}

$$\left\{ \begin{array} { l } { x \sqrt { 2 } - y \sqrt { 5 } = 2 \sqrt { 10 } } \\ { x \sqrt { 5 } + y \sqrt { 2 } = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\sqrt{5}\approx 2.236067977$$

$$y=-\sqrt{2}\approx -1.414213562$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakaraupapatia anō ngā kīanga tau.

$$\sqrt{2}x-\sqrt{5}y=2\sqrt{10}$$

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.

$$\sqrt{2}x+\left(-\sqrt{5}\right)y=2\sqrt{10},\sqrt{5}x+\sqrt{2}y=3$$

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

$$\sqrt{2}x+\left(-\sqrt{5}\right)y=2\sqrt{10}$$

Me tāpiri \sqrt{5}y ki ngā taha e rua o te whārite.

$$\sqrt{2}x=\sqrt{5}y+2\sqrt{10}$$

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

$$x=\frac{\sqrt{2}}{2}\left(\sqrt{5}y+2\sqrt{10}\right)$$

Whakareatia \frac{\sqrt{2}}{2} ki te \sqrt{5}y+2\sqrt{10}.

$$x=\frac{\sqrt{10}}{2}y+2\sqrt{5}$$

Whakakapia te \frac{\sqrt{10}y}{2}+2\sqrt{5} mō te x ki tērā atu whārite, \sqrt{5}x+\sqrt{2}y=3.

$$\sqrt{5}\left(\frac{\sqrt{10}}{2}y+2\sqrt{5}\right)+\sqrt{2}y=3$$

Whakareatia \sqrt{5} ki te \frac{\sqrt{10}y}{2}+2\sqrt{5}.

$$\frac{5\sqrt{2}}{2}y+10+\sqrt{2}y=3$$

Tāpiri \frac{5\sqrt{2}y}{2} ki te \sqrt{2}y.

$$\frac{7\sqrt{2}}{2}y+10=3$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$\frac{7\sqrt{2}}{2}y=-7$$

Whakawehea ngā taha e rua ki te \frac{7\sqrt{2}}{2}.

$$y=-\sqrt{2}$$

Whakaurua te -\sqrt{2} mō y ki x=\frac{\sqrt{10}}{2}y+2\sqrt{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{\sqrt{10}}{2}\left(-\sqrt{2}\right)+2\sqrt{5}$$

Whakareatia \frac{\sqrt{10}}{2} ki te -\sqrt{2}.

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

Tāpiri 2\sqrt{5} ki te -\sqrt{5}.

$$x=\sqrt{5}$$

Kua oti te pūnaha te whakatau.

$$x=\sqrt{5},y=-\sqrt{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakaraupapatia anō ngā kīanga tau.

$$\sqrt{2}x-\sqrt{5}y=2\sqrt{10}$$

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.

$$\sqrt{2}x+\left(-\sqrt{5}\right)y=2\sqrt{10},\sqrt{5}x+\sqrt{2}y=3$$

Kia ōrite ai a \sqrt{2}x me \sqrt{5}x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \sqrt{5} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \sqrt{2}.

$$\sqrt{5}\sqrt{2}x+\sqrt{5}\left(-\sqrt{5}\right)y=\sqrt{5}\times 2\sqrt{10},\sqrt{2}\sqrt{5}x+\sqrt{2}\sqrt{2}y=\sqrt{2}\times 3$$

Whakarūnātia.

$$\sqrt{10}x-5y=10\sqrt{2},\sqrt{10}x+2y=3\sqrt{2}$$

Me tango \sqrt{10}x+2y=3\sqrt{2} mai i \sqrt{10}x-5y=10\sqrt{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\sqrt{10}x+\left(-\sqrt{10}\right)x-5y-2y=10\sqrt{2}-3\sqrt{2}$$

Tāpiri \sqrt{10}x ki te -\sqrt{10}x. Ka whakakore atu ngā kupu \sqrt{10}x me -\sqrt{10}x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-5y-2y=10\sqrt{2}-3\sqrt{2}$$

Tāpiri -5y ki te -2y.

$$-7y=10\sqrt{2}-3\sqrt{2}$$

Tāpiri 10\sqrt{2} ki te -3\sqrt{2}.

$$-7y=7\sqrt{2}$$

Whakawehea ngā taha e rua ki te -7.

$$y=-\sqrt{2}$$

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

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

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

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

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$\sqrt{5}x=5$$

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

$$x=\sqrt{5}$$

Kua oti te pūnaha te whakatau.

$$x=\sqrt{5},y=-\sqrt{2}$$