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

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

Whakaoti mō y, x

$$x=-2$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te \frac{1}{2}x mai i ngā taha e rua.

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

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

$$y-2x=1$$

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.

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

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

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

Me tāpiri \frac{x}{2} ki ngā taha e rua o te whārite.

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

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

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

Tāpiri \frac{x}{2} ki te -2x.

$$-\frac{3}{2}x-2=1$$

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

$$-\frac{3}{2}x=3$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-2$$

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

$$y=\frac{1}{2}\left(-2\right)-2$$

Whakareatia \frac{1}{2} ki te -2.

$$y=-1-2$$

Tāpiri -2 ki te -1.

$$y=-3$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te \frac{1}{2}x mai i ngā taha e rua.

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

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

$$y-2x=1$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

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. Tangohia te \frac{1}{2}x mai i ngā taha e rua.

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

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

$$y-2x=1$$

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.

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

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

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

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.

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

Tāpiri -\frac{x}{2} ki te 2x.

$$\frac{3}{2}x=-2-1$$

Tāpiri -2 ki te -1.

$$\frac{3}{2}x=-3$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-2$$

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.

$$y-2\left(-2\right)=1$$

Whakareatia -2 ki te -2.

$$y+4=1$$

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

$$y=-3$$

Kua oti te pūnaha te whakatau.

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