Whakaoti i te {l}{3x+2y=-10}{2x-10y=-1}

$$\left. \begin{array} { l } { 3 x + 2 y = - 10 } \\ { 2 x - 10 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=-\frac{1}{2}=-0.5$$

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.

$$3x+2y=-10,2x-10y=-1$$

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

$$3x+2y=-10$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$3x=-2y-10$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y-10\right)$$

Whakareatia \frac{1}{3} ki te -2y-10.

$$x=-\frac{2}{3}y-\frac{10}{3}$$

Whakakapia te \frac{-2y-10}{3} mō te x ki tērā atu whārite, 2x-10y=-1.

$$2\left(-\frac{2}{3}y-\frac{10}{3}\right)-10y=-1$$

Whakareatia 2 ki te \frac{-2y-10}{3}.

$$-\frac{4}{3}y-\frac{20}{3}-10y=-1$$

Tāpiri -\frac{4y}{3} ki te -10y.

$$-\frac{34}{3}y-\frac{20}{3}=-1$$

Me tāpiri \frac{20}{3} ki ngā taha e rua o te whārite.

$$-\frac{34}{3}y=\frac{17}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{34}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{1}{2}$$

Whakaurua te -\frac{1}{2} mō y ki x=-\frac{2}{3}y-\frac{10}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\left(-\frac{1}{2}\right)-\frac{10}{3}$$

Whakareatia -\frac{2}{3} ki te -\frac{1}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{1-10}{3}$$

Tāpiri -\frac{10}{3} ki te \frac{1}{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.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-\frac{1}{2}$$

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.

$$3x+2y=-10,2x-10y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\2&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&2\\2&-10\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&2\\2&-10\end{matrix}\right))\left(\begin{matrix}3&2\\2&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&-10\end{matrix}\right))\left(\begin{matrix}-10\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&-10\end{matrix}\right))\left(\begin{matrix}-10\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&-10\end{matrix}\right))\left(\begin{matrix}-10\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{3\left(-10\right)-2\times 2}&-\frac{2}{3\left(-10\right)-2\times 2}\\-\frac{2}{3\left(-10\right)-2\times 2}&\frac{3}{3\left(-10\right)-2\times 2}\end{matrix}\right)\left(\begin{matrix}-10\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{17}&\frac{1}{17}\\\frac{1}{17}&-\frac{3}{34}\end{matrix}\right)\left(\begin{matrix}-10\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{17}\left(-10\right)+\frac{1}{17}\left(-1\right)\\\frac{1}{17}\left(-10\right)-\frac{3}{34}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=-\frac{1}{2}$$

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.

$$3x+2y=-10,2x-10y=-1$$

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

$$2\times 3x+2\times 2y=2\left(-10\right),3\times 2x+3\left(-10\right)y=3\left(-1\right)$$

Whakarūnātia.

$$6x+4y=-20,6x-30y=-3$$

Me tango 6x-30y=-3 mai i 6x+4y=-20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+4y+30y=-20+3$$

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

$$4y+30y=-20+3$$

Tāpiri 4y ki te 30y.

$$34y=-20+3$$

Tāpiri -20 ki te 3.

$$34y=-17$$

Whakawehea ngā taha e rua ki te 34.

$$y=-\frac{1}{2}$$

Whakaurua te -\frac{1}{2} mō y ki 2x-10y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-10\left(-\frac{1}{2}\right)=-1$$

Whakareatia -10 ki te -\frac{1}{2}.

$$2x+5=-1$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$2x=-6$$

Whakawehea ngā taha e rua ki te 2.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-\frac{1}{2}$$