Whakaoti i te {l}{-3x+y=8}{-8x+2y=20}

$$\left. \begin{array} { l } { - 3 x + y = 8 } \\ { - 8 x + 2 y = 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$-3x+y=8,-8x+2y=20$$

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

$$-3x+y=8$$

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

$$-3x=-y+8$$

Whakawehea ngā taha e rua ki te -3.

$$x=-\frac{1}{3}\left(-y+8\right)$$

Whakareatia -\frac{1}{3} ki te -y+8.

$$x=\frac{1}{3}y-\frac{8}{3}$$

Whakakapia te \frac{-8+y}{3} mō te x ki tērā atu whārite, -8x+2y=20.

$$-8\left(\frac{1}{3}y-\frac{8}{3}\right)+2y=20$$

Whakareatia -8 ki te \frac{-8+y}{3}.

$$-\frac{8}{3}y+\frac{64}{3}+2y=20$$

Tāpiri -\frac{8y}{3} ki te 2y.

$$-\frac{2}{3}y+\frac{64}{3}=20$$

Me tango \frac{64}{3} mai i ngā taha e rua o te whārite.

$$-\frac{2}{3}y=-\frac{4}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=\frac{1}{3}y-\frac{8}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\times 2-\frac{8}{3}$$

Whakareatia \frac{1}{3} ki te 2.

$$x=\frac{2-8}{3}$$

Tāpiri -\frac{8}{3} ki te \frac{2}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$-3x+y=8,-8x+2y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&1\\-8&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-3&1\\-8&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-3&1\\-8&2\end{matrix}\right))\left(\begin{matrix}-3&1\\-8&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&1\\-8&2\end{matrix}\right))\left(\begin{matrix}8\\20\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&1\\-8&2\end{matrix}\right))\left(\begin{matrix}8\\20\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&1\\-8&2\end{matrix}\right))\left(\begin{matrix}8\\20\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-3\times 2-\left(-8\right)}&-\frac{1}{-3\times 2-\left(-8\right)}\\-\frac{-8}{-3\times 2-\left(-8\right)}&-\frac{3}{-3\times 2-\left(-8\right)}\end{matrix}\right)\left(\begin{matrix}8\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&-\frac{1}{2}\\4&-\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}8\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8-\frac{1}{2}\times 20\\4\times 8-\frac{3}{2}\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=2$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$-3x+y=8,-8x+2y=20$$

Kia ōrite ai a -3x me -8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -3.

$$-8\left(-3\right)x-8y=-8\times 8,-3\left(-8\right)x-3\times 2y=-3\times 20$$

Whakarūnātia.

$$24x-8y=-64,24x-6y=-60$$

Me tango 24x-6y=-60 mai i 24x-8y=-64 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24x-24x-8y+6y=-64+60$$

Tāpiri 24x ki te -24x. Ka whakakore atu ngā kupu 24x me -24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y+6y=-64+60$$

Tāpiri -8y ki te 6y.

$$-2y=-64+60$$

Tāpiri -64 ki te 60.

$$-2y=-4$$

Whakawehea ngā taha e rua ki te -2.

$$y=2$$

Whakaurua te 2 mō y ki -8x+2y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-8x+2\times 2=20$$

Whakareatia 2 ki te 2.

$$-8x+4=20$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-8x=16$$

Whakawehea ngā taha e rua ki te -8.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=2$$