Whakaoti i te {l}{y=8x-11}{y=x+2}

$$\left. \begin{array} { l } { y = 8 x - 11 } \\ { y = x + 2 } \end{array} \right.$$

Whakaoti mō y, x

$$x = \frac{13}{7} = 1\frac{6}{7} \approx 1.857142857$$

$$y = \frac{27}{7} = 3\frac{6}{7} \approx 3.857142857$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 8x mai i ngā taha e rua.

$$y-8x=-11$$

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

$$y-x=2$$

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.

$$y-8x=-11,y-x=2$$

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

$$y-8x=-11$$

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

$$y=8x-11$$

Whakakapia te 8x-11 mō te y ki tērā atu whārite, y-x=2.

$$8x-11-x=2$$

Tāpiri 8x ki te -x.

$$7x-11=2$$

Me tāpiri 11 ki ngā taha e rua o te whārite.

$$7x=13$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{13}{7}$$

Whakaurua te \frac{13}{7} mō x ki y=8x-11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=8\times \frac{13}{7}-11$$

Whakareatia 8 ki te \frac{13}{7}.

$$y=\frac{104}{7}-11$$

Tāpiri -11 ki te \frac{104}{7}.

$$y=\frac{27}{7}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{27}{7},x=\frac{13}{7}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 8x mai i ngā taha e rua.

$$y-8x=-11$$

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

$$y-x=2$$

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

$$y-8x=-11,y-x=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-11\\2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right))\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right))\left(\begin{matrix}-11\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right))\left(\begin{matrix}-11\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right))\left(\begin{matrix}-11\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-8\right)}&-\frac{-8}{-1-\left(-8\right)}\\-\frac{1}{-1-\left(-8\right)}&\frac{1}{-1-\left(-8\right)}\end{matrix}\right)\left(\begin{matrix}-11\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}&\frac{8}{7}\\-\frac{1}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-11\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}\left(-11\right)+\frac{8}{7}\times 2\\-\frac{1}{7}\left(-11\right)+\frac{1}{7}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{27}{7}\\\frac{13}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{27}{7},x=\frac{13}{7}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 8x mai i ngā taha e rua.

$$y-8x=-11$$

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

$$y-x=2$$

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.

$$y-8x=-11,y-x=2$$

Me tango y-x=2 mai i y-8x=-11 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-8x+x=-11-2$$

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.

$$-8x+x=-11-2$$

Tāpiri -8x ki te x.

$$-7x=-11-2$$

Tāpiri -11 ki te -2.

$$-7x=-13$$

Whakawehea ngā taha e rua ki te -7.

$$x=\frac{13}{7}$$

Whakaurua te \frac{13}{7} mō x ki y-x=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-\frac{13}{7}=2$$

Me tāpiri \frac{13}{7} ki ngā taha e rua o te whārite.

$$y=\frac{27}{7}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{27}{7},x=\frac{13}{7}$$