Whakaoti i te {l}{-3x+9y=27}{-5x-8y=-1}

$$\left. \begin{array} { l } { - 3 x + 9 y = 27 } \\ { - 5 x - 8 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=2$$

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.

$$-3x+9y=27,-5x-8y=-1$$

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

$$-3x+9y=27$$

Me tango 9y mai i ngā taha e rua o te whārite.

$$-3x=-9y+27$$

Whakawehea ngā taha e rua ki te -3.

$$x=-\frac{1}{3}\left(-9y+27\right)$$

Whakareatia -\frac{1}{3} ki te -9y+27.

$$x=3y-9$$

Whakakapia te -9+3y mō te x ki tērā atu whārite, -5x-8y=-1.

$$-5\left(3y-9\right)-8y=-1$$

Whakareatia -5 ki te -9+3y.

$$-15y+45-8y=-1$$

Tāpiri -15y ki te -8y.

$$-23y+45=-1$$

Me tango 45 mai i ngā taha e rua o te whārite.

$$-23y=-46$$

Whakawehea ngā taha e rua ki te -23.

$$y=2$$

Whakaurua te 2 mō y ki x=3y-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\times 2-9$$

Whakareatia 3 ki te 2.

$$x=6-9$$

Tāpiri -9 ki te 6.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=2$$

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.

$$-3x+9y=27,-5x-8y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&9\\-5&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}27\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-3&9\\-5&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}-3&9\\-5&-8\end{matrix}\right))\left(\begin{matrix}-3&9\\-5&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&9\\-5&-8\end{matrix}\right))\left(\begin{matrix}27\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&9\\-5&-8\end{matrix}\right))\left(\begin{matrix}27\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&9\\-5&-8\end{matrix}\right))\left(\begin{matrix}27\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{-3\left(-8\right)-9\left(-5\right)}&-\frac{9}{-3\left(-8\right)-9\left(-5\right)}\\-\frac{-5}{-3\left(-8\right)-9\left(-5\right)}&-\frac{3}{-3\left(-8\right)-9\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}27\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{69}&-\frac{3}{23}\\\frac{5}{69}&-\frac{1}{23}\end{matrix}\right)\left(\begin{matrix}27\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{69}\times 27-\frac{3}{23}\left(-1\right)\\\frac{5}{69}\times 27-\frac{1}{23}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=2$$

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.

$$-3x+9y=27,-5x-8y=-1$$

Kia ōrite ai a -3x 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 -3.

$$-5\left(-3\right)x-5\times 9y=-5\times 27,-3\left(-5\right)x-3\left(-8\right)y=-3\left(-1\right)$$

Whakarūnātia.

$$15x-45y=-135,15x+24y=3$$

Me tango 15x+24y=3 mai i 15x-45y=-135 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-45y-24y=-135-3$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-45y-24y=-135-3$$

Tāpiri -45y ki te -24y.

$$-69y=-135-3$$

Tāpiri -135 ki te -3.

$$-69y=-138$$

Whakawehea ngā taha e rua ki te -69.

$$y=2$$

Whakaurua te 2 mō y ki -5x-8y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x-8\times 2=-1$$

Whakareatia -8 ki te 2.

$$-5x-16=-1$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$-5x=15$$

Whakawehea ngā taha e rua ki te -5.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=2$$