Whakaoti i te {l}{x+y=27}{y=x-9}

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

Whakaoti mō x, y

$$x=18$$

$$y=9$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te x mai i ngā taha e rua.

$$y-x=-9$$

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.

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

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

$$x+y=27$$

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

$$x=-y+27$$

Whakakapia te -y+27 mō te x ki tērā atu whārite, -x+y=-9.

$$-\left(-y+27\right)+y=-9$$

Whakareatia -1 ki te -y+27.

$$y-27+y=-9$$

Tāpiri y ki te y.

$$2y-27=-9$$

Me tāpiri 27 ki ngā taha e rua o te whārite.

$$2y=18$$

Whakawehea ngā taha e rua ki te 2.

$$y=9$$

Whakaurua te 9 mō y ki x=-y+27. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-9+27$$

Tāpiri 27 ki te -9.

$$x=18$$

Kua oti te pūnaha te whakatau.

$$x=18,y=9$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te x mai i ngā taha e rua.

$$y-x=-9$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}27\\-9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 27-\frac{1}{2}\left(-9\right)\\\frac{1}{2}\times 27+\frac{1}{2}\left(-9\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=18,y=9$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te x mai i ngā taha e rua.

$$y-x=-9$$

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.

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

Me tango -x+y=-9 mai i x+y=27 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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

$$x+x=27+9$$

Tāpiri x ki te x.

$$2x=27+9$$

Tāpiri 27 ki te 9.

$$2x=36$$

Whakawehea ngā taha e rua ki te 2.

$$x=18$$

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

$$-18+y=-9$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$y=9$$

Kua oti te pūnaha te whakatau.

$$x=18,y=9$$