Whakaoti i te {l}{y=4x-2}{y=1/4x-2}

$$\left. \begin{array} { l } { y = 4 x - 2 } \\ { y = \frac { 1 } { 4 } x - 2 } \end{array} \right.$$

Whakaoti mō y, x

$$x=0$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=-2$$

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{4}x mai i ngā taha e rua.

$$y-\frac{1}{4}x=-2$$

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.

$$y-4x=-2,y-\frac{1}{4}x=-2$$

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

$$y-4x=-2$$

Me tāpiri 4x ki ngā taha e rua o te whārite.

$$y=4x-2$$

Whakakapia te 4x-2 mō te y ki tērā atu whārite, y-\frac{1}{4}x=-2.

$$4x-2-\frac{1}{4}x=-2$$

Tāpiri 4x ki te -\frac{x}{4}.

$$\frac{15}{4}x-2=-2$$

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

$$\frac{15}{4}x=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{15}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=0$$

Whakaurua te 0 mō x ki y=4x-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-2$$

Kua oti te pūnaha te whakatau.

$$y=-2,x=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=-2$$

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{4}x mai i ngā taha e rua.

$$y-\frac{1}{4}x=-2$$

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

$$y-4x=-2,y-\frac{1}{4}x=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-4\\1&-\frac{1}{4}\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{15}&\frac{16}{15}\\-\frac{4}{15}&\frac{4}{15}\end{matrix}\right)\left(\begin{matrix}-2\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{15}\left(-2\right)+\frac{16}{15}\left(-2\right)\\-\frac{4}{15}\left(-2\right)+\frac{4}{15}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-2\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-2,x=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=-2$$

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{4}x mai i ngā taha e rua.

$$y-\frac{1}{4}x=-2$$

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.

$$y-4x=-2,y-\frac{1}{4}x=-2$$

Me tango y-\frac{1}{4}x=-2 mai i y-4x=-2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-4x+\frac{1}{4}x=-2+2$$

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.

$$-4x+\frac{1}{4}x=-2+2$$

Tāpiri -4x ki te \frac{x}{4}.

$$-\frac{15}{4}x=-2+2$$

Tāpiri -2 ki te 2.

$$-\frac{15}{4}x=0$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{15}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=0$$

Whakaurua te 0 mō x ki y-\frac{1}{4}x=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-2$$

Kua oti te pūnaha te whakatau.

$$y=-2,x=0$$