Whakaoti i te {l}{-3x+y=10}{-2x+4y=0}

$$\left. \begin{array} { l } { - 3 x + y = 10 } \\ { - 2 x + 4 y = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=-2$$

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.

$$-3x+y=10,-2x+4y=0$$

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

$$-3x+y=10$$

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

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

Whakawehea ngā taha e rua ki te -3.

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

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

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

Whakakapia te \frac{-10+y}{3} mō te x ki tērā atu whārite, -2x+4y=0.

$$-2\left(\frac{1}{3}y-\frac{10}{3}\right)+4y=0$$

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

$$-\frac{2}{3}y+\frac{20}{3}+4y=0$$

Tāpiri -\frac{2y}{3} ki te 4y.

$$\frac{10}{3}y+\frac{20}{3}=0$$

Me tango \frac{20}{3} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te \frac{10}{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{10}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

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

Tāpiri -\frac{10}{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.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-2$$

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.

$$-3x+y=10,-2x+4y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&1\\-2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-3&1\\-2&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}-3&1\\-2&4\end{matrix}\right))\left(\begin{matrix}-3&1\\-2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&1\\-2&4\end{matrix}\right))\left(\begin{matrix}10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&1\\-2&4\end{matrix}\right))\left(\begin{matrix}10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&1\\-2&4\end{matrix}\right))\left(\begin{matrix}10\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{-3\times 4-\left(-2\right)}&-\frac{1}{-3\times 4-\left(-2\right)}\\-\frac{-2}{-3\times 4-\left(-2\right)}&-\frac{3}{-3\times 4-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}10\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}&\frac{1}{10}\\-\frac{1}{5}&\frac{3}{10}\end{matrix}\right)\left(\begin{matrix}10\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}\times 10\\-\frac{1}{5}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=-2$$

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.

$$-3x+y=10,-2x+4y=0$$

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.

$$-2\left(-3\right)x-2y=-2\times 10,-3\left(-2\right)x-3\times 4y=0$$

Whakarūnātia.

$$6x-2y=-20,6x-12y=0$$

Me tango 6x-12y=0 mai i 6x-2y=-20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-2y+12y=-20$$

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.

$$-2y+12y=-20$$

Tāpiri -2y ki te 12y.

$$10y=-20$$

Whakawehea ngā taha e rua ki te 10.

$$y=-2$$

Whakaurua te -2 mō y ki -2x+4y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+4\left(-2\right)=0$$

Whakareatia 4 ki te -2.

$$-2x-8=0$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$-2x=8$$

Whakawehea ngā taha e rua ki te -2.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-2$$