Whakaoti i te {c}{4x-y=-9}{2x+2y=-2}

$$\left\{ \begin{array} { c } { 4 x - y = - 9 } \\ { 2 x + 2 y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=1$$

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.

$$4x-y=-9,2x+2y=-2$$

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

$$4x-y=-9$$

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

$$4x=y-9$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(y-9\right)$$

Whakareatia \frac{1}{4} ki te y-9.

$$x=\frac{1}{4}y-\frac{9}{4}$$

Whakakapia te \frac{-9+y}{4} mō te x ki tērā atu whārite, 2x+2y=-2.

$$2\left(\frac{1}{4}y-\frac{9}{4}\right)+2y=-2$$

Whakareatia 2 ki te \frac{-9+y}{4}.

$$\frac{1}{2}y-\frac{9}{2}+2y=-2$$

Tāpiri \frac{y}{2} ki te 2y.

$$\frac{5}{2}y-\frac{9}{2}=-2$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$\frac{5}{2}y=\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{1}{4}y-\frac{9}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1-9}{4}$$

Tāpiri -\frac{9}{4} ki te \frac{1}{4} 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.

$$x=-2,y=1$$

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.

$$4x-y=-9,2x+2y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-1\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\-2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&-1\\2&2\end{matrix}\right))\left(\begin{matrix}4&-1\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\2&2\end{matrix}\right))\left(\begin{matrix}-9\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\2&2\end{matrix}\right))\left(\begin{matrix}-9\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\2&2\end{matrix}\right))\left(\begin{matrix}-9\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{4\times 2-\left(-2\right)}&-\frac{-1}{4\times 2-\left(-2\right)}\\-\frac{2}{4\times 2-\left(-2\right)}&\frac{4}{4\times 2-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}-9\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{1}{10}\\-\frac{1}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}-9\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\left(-9\right)+\frac{1}{10}\left(-2\right)\\-\frac{1}{5}\left(-9\right)+\frac{2}{5}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=1$$

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.

$$4x-y=-9,2x+2y=-2$$

Kia ōrite ai a 4x 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 4.

$$2\times 4x+2\left(-1\right)y=2\left(-9\right),4\times 2x+4\times 2y=4\left(-2\right)$$

Whakarūnātia.

$$8x-2y=-18,8x+8y=-8$$

Me tango 8x+8y=-8 mai i 8x-2y=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-2y-8y=-18+8$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-8y=-18+8$$

Tāpiri -2y ki te -8y.

$$-10y=-18+8$$

Tāpiri -18 ki te 8.

$$-10y=-10$$

Whakawehea ngā taha e rua ki te -10.

$$y=1$$

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

$$2x+2=-2$$

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

$$2x=-4$$

Whakawehea ngā taha e rua ki te 2.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=1$$