Whakaoti i te {l}{9x+y=88}{7x-8y=7}

$$\left. \begin{array} { l } { 9 x + y = 88 } \\ { 7 x - 8 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=7$$

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.

$$9x+y=88,7x-8y=7$$

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

$$9x+y=88$$

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

$$9x=-y+88$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(-y+88\right)$$

Whakareatia \frac{1}{9} ki te -y+88.

$$x=-\frac{1}{9}y+\frac{88}{9}$$

Whakakapia te \frac{-y+88}{9} mō te x ki tērā atu whārite, 7x-8y=7.

$$7\left(-\frac{1}{9}y+\frac{88}{9}\right)-8y=7$$

Whakareatia 7 ki te \frac{-y+88}{9}.

$$-\frac{7}{9}y+\frac{616}{9}-8y=7$$

Tāpiri -\frac{7y}{9} ki te -8y.

$$-\frac{79}{9}y+\frac{616}{9}=7$$

Me tango \frac{616}{9} mai i ngā taha e rua o te whārite.

$$-\frac{79}{9}y=-\frac{553}{9}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{79}{9}, 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{1}{9}y+\frac{88}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{9}\times 7+\frac{88}{9}$$

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

$$x=\frac{-7+88}{9}$$

Tāpiri \frac{88}{9} ki te -\frac{7}{9} 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.

$$x=9,y=7$$

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.

$$9x+y=88,7x-8y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&1\\7&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}88\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&1\\7&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&1\\7&-8\end{matrix}\right))\left(\begin{matrix}9&1\\7&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&1\\7&-8\end{matrix}\right))\left(\begin{matrix}88\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&1\\7&-8\end{matrix}\right))\left(\begin{matrix}88\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&1\\7&-8\end{matrix}\right))\left(\begin{matrix}88\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{9\left(-8\right)-7}&-\frac{1}{9\left(-8\right)-7}\\-\frac{7}{9\left(-8\right)-7}&\frac{9}{9\left(-8\right)-7}\end{matrix}\right)\left(\begin{matrix}88\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{79}&\frac{1}{79}\\\frac{7}{79}&-\frac{9}{79}\end{matrix}\right)\left(\begin{matrix}88\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{79}\times 88+\frac{1}{79}\times 7\\\frac{7}{79}\times 88-\frac{9}{79}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=9,y=7$$

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.

$$9x+y=88,7x-8y=7$$

Kia ōrite ai a 9x 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 9.

$$7\times 9x+7y=7\times 88,9\times 7x+9\left(-8\right)y=9\times 7$$

Whakarūnātia.

$$63x+7y=616,63x-72y=63$$

Me tango 63x-72y=63 mai i 63x+7y=616 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$63x-63x+7y+72y=616-63$$

Tāpiri 63x ki te -63x. Ka whakakore atu ngā kupu 63x me -63x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7y+72y=616-63$$

Tāpiri 7y ki te 72y.

$$79y=616-63$$

Tāpiri 616 ki te -63.

$$79y=553$$

Whakawehea ngā taha e rua ki te 79.

$$y=7$$

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

$$7x-8\times 7=7$$

Whakareatia -8 ki te 7.

$$7x-56=7$$

Me tāpiri 56 ki 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.

$$x=9,y=7$$