Whakaoti i te {l}{y=6x+4}{y=8x+2}

$$\left. \begin{array} { l } { y = 6 x + 4 } \\ { y = 8 x + 2 } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=10$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-6x=4$$

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

$$y-8x=2$$

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.

$$y-6x=4,y-8x=2$$

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

$$y-6x=4$$

Me tāpiri 6x ki ngā taha e rua o te whārite.

$$y=6x+4$$

Whakakapia te 6x+4 mō te y ki tērā atu whārite, y-8x=2.

$$6x+4-8x=2$$

Tāpiri 6x ki te -8x.

$$-2x+4=2$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-2x=-2$$

Whakawehea ngā taha e rua ki te -2.

$$x=1$$

Whakaurua te 1 mō x ki y=6x+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=6+4$$

Tāpiri 4 ki te 6.

$$y=10$$

Kua oti te pūnaha te whakatau.

$$y=10,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-6x=4$$

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

$$y-8x=2$$

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

$$y-6x=4,y-8x=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-6\\1&-8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}4\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-6\\1&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-6\\1&-8\end{matrix}\right))\left(\begin{matrix}1&-6\\1&-8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-6\\1&-8\end{matrix}\right))\left(\begin{matrix}4\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-6\\1&-8\end{matrix}\right))\left(\begin{matrix}4\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-6\\1&-8\end{matrix}\right))\left(\begin{matrix}4\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{-8-\left(-6\right)}&-\frac{-6}{-8-\left(-6\right)}\\-\frac{1}{-8-\left(-6\right)}&\frac{1}{-8-\left(-6\right)}\end{matrix}\right)\left(\begin{matrix}4\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}4&-3\\\frac{1}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}4\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}4\times 4-3\times 2\\\frac{1}{2}\times 4-\frac{1}{2}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}10\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=10,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-6x=4$$

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

$$y-8x=2$$

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.

$$y-6x=4,y-8x=2$$

Me tango y-8x=2 mai i y-6x=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-6x+8x=4-2$$

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.

$$-6x+8x=4-2$$

Tāpiri -6x ki te 8x.

$$2x=4-2$$

Tāpiri 4 ki te -2.

$$2x=2$$

Whakawehea ngā taha e rua ki te 2.

$$x=1$$

Whakaurua te 1 mō x ki y-8x=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-8=2$$

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

$$y=10$$

Kua oti te pūnaha te whakatau.

$$y=10,x=1$$