Whakaoti i te {l}{14x-3y=-63}{7x+2y=-7}

$$\left\{ \begin{array} { l } { 14 x - 3 y = - 63 } \\ { 7 x + 2 y = - 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=7$$

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.

$$14x-3y=-63,7x+2y=-7$$

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

$$14x-3y=-63$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$14x=3y-63$$

Whakawehea ngā taha e rua ki te 14.

$$x=\frac{1}{14}\left(3y-63\right)$$

Whakareatia \frac{1}{14} ki te -63+3y.

$$x=\frac{3}{14}y-\frac{9}{2}$$

Whakakapia te \frac{3y}{14}-\frac{9}{2} mō te x ki tērā atu whārite, 7x+2y=-7.

$$7\left(\frac{3}{14}y-\frac{9}{2}\right)+2y=-7$$

Whakareatia 7 ki te \frac{3y}{14}-\frac{9}{2}.

$$\frac{3}{2}y-\frac{63}{2}+2y=-7$$

Tāpiri \frac{3y}{2} ki te 2y.

$$\frac{7}{2}y-\frac{63}{2}=-7$$

Me tāpiri \frac{63}{2} ki ngā taha e rua o te whārite.

$$\frac{7}{2}y=\frac{49}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=\frac{3}{14}y-\frac{9}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{14}\times 7-\frac{9}{2}$$

Whakareatia \frac{3}{14} ki te 7.

$$x=\frac{3-9}{2}$$

Tāpiri -\frac{9}{2} ki te \frac{3}{2} 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.

$$x=-3,y=7$$

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.

$$14x-3y=-63,7x+2y=-7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}14&-3\\7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-63\\-7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}14&-3\\7&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}14&-3\\7&2\end{matrix}\right))\left(\begin{matrix}14&-3\\7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}14&-3\\7&2\end{matrix}\right))\left(\begin{matrix}-63\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}14&-3\\7&2\end{matrix}\right))\left(\begin{matrix}-63\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}14&-3\\7&2\end{matrix}\right))\left(\begin{matrix}-63\\-7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{14\times 2-\left(-3\times 7\right)}&-\frac{-3}{14\times 2-\left(-3\times 7\right)}\\-\frac{7}{14\times 2-\left(-3\times 7\right)}&\frac{14}{14\times 2-\left(-3\times 7\right)}\end{matrix}\right)\left(\begin{matrix}-63\\-7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{49}&\frac{3}{49}\\-\frac{1}{7}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}-63\\-7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{49}\left(-63\right)+\frac{3}{49}\left(-7\right)\\-\frac{1}{7}\left(-63\right)+\frac{2}{7}\left(-7\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=7$$

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.

$$14x-3y=-63,7x+2y=-7$$

Kia ōrite ai a 14x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 14.

$$7\times 14x+7\left(-3\right)y=7\left(-63\right),14\times 7x+14\times 2y=14\left(-7\right)$$

Whakarūnātia.

$$98x-21y=-441,98x+28y=-98$$

Me tango 98x+28y=-98 mai i 98x-21y=-441 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$98x-98x-21y-28y=-441+98$$

Tāpiri 98x ki te -98x. Ka whakakore atu ngā kupu 98x me -98x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-21y-28y=-441+98$$

Tāpiri -21y ki te -28y.

$$-49y=-441+98$$

Tāpiri -441 ki te 98.

$$-49y=-343$$

Whakawehea ngā taha e rua ki te -49.

$$y=7$$

Whakaurua te 7 mō y ki 7x+2y=-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+2\times 7=-7$$

Whakareatia 2 ki te 7.

$$7x+14=-7$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$7x=-21$$

Whakawehea ngā taha e rua ki te 7.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=7$$