Whakaoti i te {l}{-8x+7y=-9}{-9x+7y=-18}

$$\left. \begin{array} { l } { - 8 x + 7 y = - 9 } \\ { - 9 x + 7 y = - 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=9$$

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.

$$-8x+7y=-9,-9x+7y=-18$$

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

$$-8x+7y=-9$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$-8x=-7y-9$$

Whakawehea ngā taha e rua ki te -8.

$$x=-\frac{1}{8}\left(-7y-9\right)$$

Whakareatia -\frac{1}{8} ki te -7y-9.

$$x=\frac{7}{8}y+\frac{9}{8}$$

Whakakapia te \frac{7y+9}{8} mō te x ki tērā atu whārite, -9x+7y=-18.

$$-9\left(\frac{7}{8}y+\frac{9}{8}\right)+7y=-18$$

Whakareatia -9 ki te \frac{7y+9}{8}.

$$-\frac{63}{8}y-\frac{81}{8}+7y=-18$$

Tāpiri -\frac{63y}{8} ki te 7y.

$$-\frac{7}{8}y-\frac{81}{8}=-18$$

Me tāpiri \frac{81}{8} ki ngā taha e rua o te whārite.

$$-\frac{7}{8}y=-\frac{63}{8}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{8}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=9$$

Whakaurua te 9 mō y ki x=\frac{7}{8}y+\frac{9}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{8}\times 9+\frac{9}{8}$$

Whakareatia \frac{7}{8} ki te 9.

$$x=\frac{63+9}{8}$$

Tāpiri \frac{9}{8} ki te \frac{63}{8} 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.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=9$$

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.

$$-8x+7y=-9,-9x+7y=-18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-8&7\\-9&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\-18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-8&7\\-9&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}-8&7\\-9&7\end{matrix}\right))\left(\begin{matrix}-8&7\\-9&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&7\\-9&7\end{matrix}\right))\left(\begin{matrix}-9\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&7\\-9&7\end{matrix}\right))\left(\begin{matrix}-9\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&7\\-9&7\end{matrix}\right))\left(\begin{matrix}-9\\-18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{-8\times 7-7\left(-9\right)}&-\frac{7}{-8\times 7-7\left(-9\right)}\\-\frac{-9}{-8\times 7-7\left(-9\right)}&-\frac{8}{-8\times 7-7\left(-9\right)}\end{matrix}\right)\left(\begin{matrix}-9\\-18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&-1\\\frac{9}{7}&-\frac{8}{7}\end{matrix}\right)\left(\begin{matrix}-9\\-18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9-\left(-18\right)\\\frac{9}{7}\left(-9\right)-\frac{8}{7}\left(-18\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=9$$

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.

$$-8x+7y=-9,-9x+7y=-18$$

Me tango -9x+7y=-18 mai i -8x+7y=-9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+9x+7y-7y=-9+18$$

Tāpiri 7y ki te -7y. Ka whakakore atu ngā kupu 7y me -7y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8x+9x=-9+18$$

Tāpiri -8x ki te 9x.

$$x=-9+18$$

Tāpiri -9 ki te 18.

$$x=9$$

Whakaurua te 9 mō x ki -9x+7y=-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-9\times 9+7y=-18$$

Whakareatia -9 ki te 9.

$$-81+7y=-18$$

Me tāpiri 81 ki ngā taha e rua o te whārite.

$$7y=63$$

Whakawehea ngā taha e rua ki te 7.

$$y=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=9$$