Whakaoti i te {l}{y=8x-18}{y=x-4}

$$\left. \begin{array} { l } { y = 8 x - 18 } \\ { y = x - 4 } \end{array} \right.$$

Whakaoti mō y, x

$$x=2$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-8x=-18$$

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

$$y-x=-4$$

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.

$$y-8x=-18,y-x=-4$$

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

$$y-8x=-18$$

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

$$y=8x-18$$

Whakakapia te 8x-18 mō te y ki tērā atu whārite, y-x=-4.

$$8x-18-x=-4$$

Tāpiri 8x ki te -x.

$$7x-18=-4$$

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

$$7x=14$$

Whakawehea ngā taha e rua ki te 7.

$$x=2$$

Whakaurua te 2 mō x ki y=8x-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=8\times 2-18$$

Whakareatia 8 ki te 2.

$$y=16-18$$

Tāpiri -18 ki te 16.

$$y=-2$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-8x=-18$$

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

$$y-x=-4$$

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

$$y-8x=-18,y-x=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-18\\-4\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right))\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right))\left(\begin{matrix}-18\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right))\left(\begin{matrix}-18\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&-1\end{matrix}\right))\left(\begin{matrix}-18\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-8\right)}&-\frac{-8}{-1-\left(-8\right)}\\-\frac{1}{-1-\left(-8\right)}&\frac{1}{-1-\left(-8\right)}\end{matrix}\right)\left(\begin{matrix}-18\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}&\frac{8}{7}\\-\frac{1}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-18\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}\left(-18\right)+\frac{8}{7}\left(-4\right)\\-\frac{1}{7}\left(-18\right)+\frac{1}{7}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-8x=-18$$

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

$$y-x=-4$$

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.

$$y-8x=-18,y-x=-4$$

Me tango y-x=-4 mai i y-8x=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-8x+x=-18+4$$

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.

$$-8x+x=-18+4$$

Tāpiri -8x ki te x.

$$-7x=-18+4$$

Tāpiri -18 ki te 4.

$$-7x=-14$$

Whakawehea ngā taha e rua ki te -7.

$$x=2$$

Whakaurua te 2 mō x ki y-x=-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-2=-4$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$y=-2$$

Kua oti te pūnaha te whakatau.

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