Whakaoti i te {l}{x+6y=90}{3x+3y=-30}

$$\left\{ \begin{array} { l } { x + 6 y = 90 } \\ { 3 x + 3 y = - 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-30$$

$$y=20$$

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.

$$x+6y=90,3x+3y=-30$$

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

$$x+6y=90$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$x=-6y+90$$

Whakakapia te -6y+90 mō te x ki tērā atu whārite, 3x+3y=-30.

$$3\left(-6y+90\right)+3y=-30$$

Whakareatia 3 ki te -6y+90.

$$-18y+270+3y=-30$$

Tāpiri -18y ki te 3y.

$$-15y+270=-30$$

Me tango 270 mai i ngā taha e rua o te whārite.

$$-15y=-300$$

Whakawehea ngā taha e rua ki te -15.

$$y=20$$

Whakaurua te 20 mō y ki x=-6y+90. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-6\times 20+90$$

Whakareatia -6 ki te 20.

$$x=-120+90$$

Tāpiri 90 ki te -120.

$$x=-30$$

Kua oti te pūnaha te whakatau.

$$x=-30,y=20$$

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.

$$x+6y=90,3x+3y=-30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&6\\3&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}90\\-30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&6\\3&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&6\\3&3\end{matrix}\right))\left(\begin{matrix}1&6\\3&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&6\\3&3\end{matrix}\right))\left(\begin{matrix}90\\-30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&6\\3&3\end{matrix}\right))\left(\begin{matrix}90\\-30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&6\\3&3\end{matrix}\right))\left(\begin{matrix}90\\-30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-6\times 3}&-\frac{6}{3-6\times 3}\\-\frac{3}{3-6\times 3}&\frac{1}{3-6\times 3}\end{matrix}\right)\left(\begin{matrix}90\\-30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{2}{5}\\\frac{1}{5}&-\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}90\\-30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 90+\frac{2}{5}\left(-30\right)\\\frac{1}{5}\times 90-\frac{1}{15}\left(-30\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-30\\20\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-30,y=20$$

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.

$$x+6y=90,3x+3y=-30$$

Kia ōrite ai a x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$3x+3\times 6y=3\times 90,3x+3y=-30$$

Whakarūnātia.

$$3x+18y=270,3x+3y=-30$$

Me tango 3x+3y=-30 mai i 3x+18y=270 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+18y-3y=270+30$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$18y-3y=270+30$$

Tāpiri 18y ki te -3y.

$$15y=270+30$$

Tāpiri 270 ki te 30.

$$15y=300$$

Whakawehea ngā taha e rua ki te 15.

$$y=20$$

Whakaurua te 20 mō y ki 3x+3y=-30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+3\times 20=-30$$

Whakareatia 3 ki te 20.

$$3x+60=-30$$

Me tango 60 mai i ngā taha e rua o te whārite.

$$3x=-90$$

Whakawehea ngā taha e rua ki te 3.

$$x=-30$$

Kua oti te pūnaha te whakatau.

$$x=-30,y=20$$