Whakaoti i te {l}{y=-2x+1}{y=6x-15}

$$\left. \begin{array} { l } { y = - 2 x + 1 } \\ { y = 6 x - 15 } \end{array} \right.$$

Whakaoti mō y, x

$$x=2$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=1$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=-15$$

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.

$$y+2x=1,y-6x=-15$$

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

$$y+2x=1$$

Me tango 2x mai i ngā taha e rua o te whārite.

$$y=-2x+1$$

Whakakapia te -2x+1 mō te y ki tērā atu whārite, y-6x=-15.

$$-2x+1-6x=-15$$

Tāpiri -2x ki te -6x.

$$-8x+1=-15$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-8x=-16$$

Whakawehea ngā taha e rua ki te -8.

$$x=2$$

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

$$y=-2\times 2+1$$

Whakareatia -2 ki te 2.

$$y=-4+1$$

Tāpiri 1 ki te -4.

$$y=-3$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=1$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=-15$$

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

$$y+2x=1,y-6x=-15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\1&-6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1\\-15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\1&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\1&-6\end{matrix}\right))\left(\begin{matrix}1&2\\1&-6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-6\end{matrix}\right))\left(\begin{matrix}1\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-6\end{matrix}\right))\left(\begin{matrix}1\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-6\end{matrix}\right))\left(\begin{matrix}1\\-15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-6-2}&-\frac{2}{-6-2}\\-\frac{1}{-6-2}&\frac{1}{-6-2}\end{matrix}\right)\left(\begin{matrix}1\\-15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&\frac{1}{4}\\\frac{1}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}1\\-15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}+\frac{1}{4}\left(-15\right)\\\frac{1}{8}-\frac{1}{8}\left(-15\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=1$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=-15$$

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.

$$y+2x=1,y-6x=-15$$

Me tango y-6x=-15 mai i y+2x=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+2x+6x=1+15$$

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.

$$2x+6x=1+15$$

Tāpiri 2x ki te 6x.

$$8x=1+15$$

Tāpiri 1 ki te 15.

$$8x=16$$

Whakawehea ngā taha e rua ki te 8.

$$x=2$$

Whakaurua te 2 mō x ki y-6x=-15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-6\times 2=-15$$

Whakareatia -6 ki te 2.

$$y-12=-15$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$y=-3$$

Kua oti te pūnaha te whakatau.

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