Whakaoti i te {l}{y=1/2x+1}{2y+3x=-2}

$$\left. \begin{array} { l } { y = \frac { 1 } { 2 } x + 1 } \\ { 2 y + 3 x = - 2 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-1$$

$$y=\frac{1}{2}=0.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

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

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.

$$y-\frac{1}{2}x=1,2y+3x=-2$$

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

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

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

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

Whakakapia te \frac{x}{2}+1 mō te y ki tērā atu whārite, 2y+3x=-2.

$$2\left(\frac{1}{2}x+1\right)+3x=-2$$

Whakareatia 2 ki te \frac{x}{2}+1.

$$x+2+3x=-2$$

Tāpiri x ki te 3x.

$$4x+2=-2$$

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

$$4x=-4$$

Whakawehea ngā taha e rua ki te 4.

$$x=-1$$

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

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

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

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

Tāpiri 1 ki te -\frac{1}{2}.

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

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

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

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

$$y-\frac{1}{2}x=1,2y+3x=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-\frac{1}{2}\\2&3\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{2}\\2&3\end{matrix}\right))\left(\begin{matrix}1\\-2\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&\frac{1}{8}\\-\frac{1}{2}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}1\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}+\frac{1}{8}\left(-2\right)\\-\frac{1}{2}+\frac{1}{4}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

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

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

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.

$$y-\frac{1}{2}x=1,2y+3x=-2$$

Kia ōrite ai a y me 2y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2y+2\left(-\frac{1}{2}\right)x=2,2y+3x=-2$$

Whakarūnātia.

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

Me tango 2y+3x=-2 mai i 2y-x=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2y-2y-x-3x=2+2$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-x-3x=2+2$$

Tāpiri -x ki te -3x.

$$-4x=2+2$$

Tāpiri 2 ki te 2.

$$-4x=4$$

Whakawehea ngā taha e rua ki te -4.

$$x=-1$$

Whakaurua te -1 mō x ki 2y+3x=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y+3\left(-1\right)=-2$$

Whakareatia 3 ki te -1.

$$2y-3=-2$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$2y=1$$

Whakawehea ngā taha e rua ki te 2.

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

Kua oti te pūnaha te whakatau.

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