Whakaoti i te {l}{y=mx+6}{y=a+2x}

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

Whakaoti mō x, y

$$\left\{\begin{matrix}x=-\frac{6-a}{m-2}\text{, }y=-\frac{12-am}{m-2}\text{, }&m\neq 2\\x=\frac{y-6}{2}\text{, }y\in \mathrm{R}\text{, }&a=6\text{ and }m=2\end{matrix}\right.$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-mx=6$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=a$$

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.

$$y+\left(-m\right)x=6,y-2x=a$$

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

$$y+\left(-m\right)x=6$$

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

$$y=mx+6$$

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

$$mx+6-2x=a$$

Tāpiri mx ki te -2x.

$$\left(m-2\right)x+6=a$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$\left(m-2\right)x=a-6$$

Whakawehea ngā taha e rua ki te m-2.

$$x=\frac{a-6}{m-2}$$

Whakaurua te \frac{a-6}{m-2} mō x ki y=mx+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=m\times \frac{a-6}{m-2}+6$$

Whakareatia m ki te \frac{a-6}{m-2}.

$$y=\frac{m\left(a-6\right)}{m-2}+6$$

Tāpiri 6 ki te \frac{m\left(a-6\right)}{m-2}.

$$y=\frac{am-12}{m-2}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{am-12}{m-2},x=\frac{a-6}{m-2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-mx=6$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=a$$

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

$$y+\left(-m\right)x=6,y-2x=a$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\left(-\frac{2}{m-2}\right)\times 6+\frac{m}{m-2}a\\\left(-\frac{1}{m-2}\right)\times 6+\frac{1}{m-2}a\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{am-12}{m-2}\\\frac{a-6}{m-2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{am-12}{m-2},x=\frac{a-6}{m-2}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-mx=6$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=a$$

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.

$$y+\left(-m\right)x=6,y-2x=a$$

Me tango y-2x=a mai i y+\left(-m\right)x=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+\left(-m\right)x+2x=6-a$$

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.

$$\left(-m\right)x+2x=6-a$$

Tāpiri -mx ki te 2x.

$$\left(2-m\right)x=6-a$$

Whakawehea ngā taha e rua ki te -m+2.

$$x=\frac{6-a}{2-m}$$

Whakaurua te \frac{6-a}{-m+2} mō x ki y-2x=a. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-2\times \frac{6-a}{2-m}=a$$

Whakareatia -2 ki te \frac{6-a}{-m+2}.

$$y-\frac{2\left(6-a\right)}{2-m}=a$$

Me tāpiri \frac{2\left(6-a\right)}{-m+2} ki ngā taha e rua o te whārite.

$$y=\frac{12-am}{2-m}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{12-am}{2-m},x=\frac{6-a}{2-m}$$