Whakaoti i te {l}{-7x-8y=-2}{-5x+8y=26}

$$\left. \begin{array} { l } { - 7 x - 8 y = - 2 } \\ { - 5 x + 8 y = 26 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=2$$

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.

$$-7x-8y=-2,-5x+8y=26$$

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

$$-7x-8y=-2$$

Me tāpiri 8y ki ngā taha e rua o te whārite.

$$-7x=8y-2$$

Whakawehea ngā taha e rua ki te -7.

$$x=-\frac{1}{7}\left(8y-2\right)$$

Whakareatia -\frac{1}{7} ki te 8y-2.

$$x=-\frac{8}{7}y+\frac{2}{7}$$

Whakakapia te \frac{-8y+2}{7} mō te x ki tērā atu whārite, -5x+8y=26.

$$-5\left(-\frac{8}{7}y+\frac{2}{7}\right)+8y=26$$

Whakareatia -5 ki te \frac{-8y+2}{7}.

$$\frac{40}{7}y-\frac{10}{7}+8y=26$$

Tāpiri \frac{40y}{7} ki te 8y.

$$\frac{96}{7}y-\frac{10}{7}=26$$

Me tāpiri \frac{10}{7} ki ngā taha e rua o te whārite.

$$\frac{96}{7}y=\frac{192}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{96}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{8}{7}y+\frac{2}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{8}{7}\times 2+\frac{2}{7}$$

Whakareatia -\frac{8}{7} ki te 2.

$$x=\frac{-16+2}{7}$$

Tāpiri \frac{2}{7} ki te -\frac{16}{7} 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.

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

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.

$$-7x-8y=-2,-5x+8y=26$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-7&-8\\-5&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\26\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-7&-8\\-5&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}-7&-8\\-5&8\end{matrix}\right))\left(\begin{matrix}-7&-8\\-5&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-8\\-5&8\end{matrix}\right))\left(\begin{matrix}-2\\26\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-8\\-5&8\end{matrix}\right))\left(\begin{matrix}-2\\26\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-8\\-5&8\end{matrix}\right))\left(\begin{matrix}-2\\26\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{-7\times 8-\left(-8\left(-5\right)\right)}&-\frac{-8}{-7\times 8-\left(-8\left(-5\right)\right)}\\-\frac{-5}{-7\times 8-\left(-8\left(-5\right)\right)}&-\frac{7}{-7\times 8-\left(-8\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}-2\\26\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}&-\frac{1}{12}\\-\frac{5}{96}&\frac{7}{96}\end{matrix}\right)\left(\begin{matrix}-2\\26\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}\left(-2\right)-\frac{1}{12}\times 26\\-\frac{5}{96}\left(-2\right)+\frac{7}{96}\times 26\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$-7x-8y=-2,-5x+8y=26$$

Kia ōrite ai a -7x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -7.

$$-5\left(-7\right)x-5\left(-8\right)y=-5\left(-2\right),-7\left(-5\right)x-7\times 8y=-7\times 26$$

Whakarūnātia.

$$35x+40y=10,35x-56y=-182$$

Me tango 35x-56y=-182 mai i 35x+40y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$35x-35x+40y+56y=10+182$$

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.

$$40y+56y=10+182$$

Tāpiri 40y ki te 56y.

$$96y=10+182$$

Tāpiri 10 ki te 182.

$$96y=192$$

Whakawehea ngā taha e rua ki te 96.

$$y=2$$

Whakaurua te 2 mō y ki -5x+8y=26. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+8\times 2=26$$

Whakareatia 8 ki te 2.

$$-5x+16=26$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$-5x=10$$

Whakawehea ngā taha e rua ki te -5.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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