Whakaoti i te {l}{-8x+7y=13}{7x-9y=-20}

$$\left. \begin{array} { l } { - 8 x + 7 y = 13 } \\ { 7 x - 9 y = - 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=3$$

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.

$$-8x+7y=13,7x-9y=-20$$

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

$$-8x+7y=13$$

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

$$-8x=-7y+13$$

Whakawehea ngā taha e rua ki te -8.

$$x=-\frac{1}{8}\left(-7y+13\right)$$

Whakareatia -\frac{1}{8} ki te -7y+13.

$$x=\frac{7}{8}y-\frac{13}{8}$$

Whakakapia te \frac{7y-13}{8} mō te x ki tērā atu whārite, 7x-9y=-20.

$$7\left(\frac{7}{8}y-\frac{13}{8}\right)-9y=-20$$

Whakareatia 7 ki te \frac{7y-13}{8}.

$$\frac{49}{8}y-\frac{91}{8}-9y=-20$$

Tāpiri \frac{49y}{8} ki te -9y.

$$-\frac{23}{8}y-\frac{91}{8}=-20$$

Me tāpiri \frac{91}{8} ki ngā taha e rua o te whārite.

$$-\frac{23}{8}y=-\frac{69}{8}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{23}{8}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=\frac{7}{8}y-\frac{13}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{8}\times 3-\frac{13}{8}$$

Whakareatia \frac{7}{8} ki te 3.

$$x=\frac{21-13}{8}$$

Tāpiri -\frac{13}{8} ki te \frac{21}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=3$$

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.

$$-8x+7y=13,7x-9y=-20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-8&7\\7&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\-20\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-8&7\\7&-9\end{matrix}\right))\left(\begin{matrix}-8&7\\7&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&7\\7&-9\end{matrix}\right))\left(\begin{matrix}13\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&7\\7&-9\end{matrix}\right))\left(\begin{matrix}13\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&7\\7&-9\end{matrix}\right))\left(\begin{matrix}13\\-20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{-8\left(-9\right)-7\times 7}&-\frac{7}{-8\left(-9\right)-7\times 7}\\-\frac{7}{-8\left(-9\right)-7\times 7}&-\frac{8}{-8\left(-9\right)-7\times 7}\end{matrix}\right)\left(\begin{matrix}13\\-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{23}&-\frac{7}{23}\\-\frac{7}{23}&-\frac{8}{23}\end{matrix}\right)\left(\begin{matrix}13\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{23}\times 13-\frac{7}{23}\left(-20\right)\\-\frac{7}{23}\times 13-\frac{8}{23}\left(-20\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=3$$

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.

$$-8x+7y=13,7x-9y=-20$$

Kia ōrite ai a -8x 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 -8.

$$7\left(-8\right)x+7\times 7y=7\times 13,-8\times 7x-8\left(-9\right)y=-8\left(-20\right)$$

Whakarūnātia.

$$-56x+49y=91,-56x+72y=160$$

Me tango -56x+72y=160 mai i -56x+49y=91 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-56x+56x+49y-72y=91-160$$

Tāpiri -56x ki te 56x. Ka whakakore atu ngā kupu -56x me 56x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$49y-72y=91-160$$

Tāpiri 49y ki te -72y.

$$-23y=91-160$$

Tāpiri 91 ki te -160.

$$-23y=-69$$

Whakawehea ngā taha e rua ki te -23.

$$y=3$$

Whakaurua te 3 mō y ki 7x-9y=-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x-9\times 3=-20$$

Whakareatia -9 ki te 3.

$$7x-27=-20$$

Me tāpiri 27 ki ngā taha e rua o te whārite.

$$7x=7$$

Whakawehea ngā taha e rua ki te 7.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=3$$