Whakaoti i te {l}{-5x-7y=-18}{7x+9y=18}

$$\left. \begin{array} { l } { - 5 x - 7 y = - 18 } \\ { 7 x + 9 y = 18 } \end{array} \right.$$

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.

$$-5x-7y=-18,7x+9y=18$$

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

$$-5x-7y=-18$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$-5x=7y-18$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{1}{5}\left(7y-18\right)$$

Whakareatia -\frac{1}{5} ki te 7y-18.

$$x=-\frac{7}{5}y+\frac{18}{5}$$

Whakakapia te \frac{-7y+18}{5} mō te x ki tērā atu whārite, 7x+9y=18.

$$7\left(-\frac{7}{5}y+\frac{18}{5}\right)+9y=18$$

Whakareatia 7 ki te \frac{-7y+18}{5}.

$$-\frac{49}{5}y+\frac{126}{5}+9y=18$$

Tāpiri -\frac{49y}{5} ki te 9y.

$$-\frac{4}{5}y+\frac{126}{5}=18$$

Me tango \frac{126}{5} mai i ngā taha e rua o te whārite.

$$-\frac{4}{5}y=-\frac{36}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{4}{5}, 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}{5}y+\frac{18}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{5}\times 9+\frac{18}{5}$$

Whakareatia -\frac{7}{5} ki te 9.

$$x=\frac{-63+18}{5}$$

Tāpiri \frac{18}{5} ki te -\frac{63}{5} 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.

$$-5x-7y=-18,7x+9y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&-7\\7&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-18\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&-7\\7&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&-7\\7&9\end{matrix}\right))\left(\begin{matrix}-5&-7\\7&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-7\\7&9\end{matrix}\right))\left(\begin{matrix}-18\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-7\\7&9\end{matrix}\right))\left(\begin{matrix}-18\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-7\\7&9\end{matrix}\right))\left(\begin{matrix}-18\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{-5\times 9-\left(-7\times 7\right)}&-\frac{-7}{-5\times 9-\left(-7\times 7\right)}\\-\frac{7}{-5\times 9-\left(-7\times 7\right)}&-\frac{5}{-5\times 9-\left(-7\times 7\right)}\end{matrix}\right)\left(\begin{matrix}-18\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{4}&\frac{7}{4}\\-\frac{7}{4}&-\frac{5}{4}\end{matrix}\right)\left(\begin{matrix}-18\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{4}\left(-18\right)+\frac{7}{4}\times 18\\-\frac{7}{4}\left(-18\right)-\frac{5}{4}\times 18\end{matrix}\right)$$

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.

$$-5x-7y=-18,7x+9y=18$$

Kia ōrite ai a -5x 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 -5.

$$7\left(-5\right)x+7\left(-7\right)y=7\left(-18\right),-5\times 7x-5\times 9y=-5\times 18$$

Whakarūnātia.

$$-35x-49y=-126,-35x-45y=-90$$

Me tango -35x-45y=-90 mai i -35x-49y=-126 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-35x+35x-49y+45y=-126+90$$

Tāpiri -35x ki te 35x. Ka whakakore atu ngā kupu -35x me 35x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-49y+45y=-126+90$$

Tāpiri -49y ki te 45y.

$$-4y=-126+90$$

Tāpiri -126 ki te 90.

$$-4y=-36$$

Whakawehea ngā taha e rua ki te -4.

$$y=9$$

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

$$7x+9\times 9=18$$

Whakareatia 9 ki te 9.

$$7x+81=18$$

Me tango 81 mai i ngā taha e rua o te whārite.

$$7x=-63$$

Whakawehea ngā taha e rua ki te 7.

$$x=-9$$

Kua oti te pūnaha te whakatau.

$$x=-9,y=9$$