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

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

Whakaoti mō y, x

$$x=-\frac{1}{2}=-0.5$$

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

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-x=1$$

Whakaarohia te whārite tuarua. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=2$$

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.

$$y-x=1,y-3x=2$$

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

$$y-x=1$$

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

$$y=x+1$$

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

$$x+1-3x=2$$

Tāpiri x ki te -3x.

$$-2x+1=2$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-2x=1$$

Whakawehea ngā taha e rua ki te -2.

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

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

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

Tāpiri 1 ki te -\frac{1}{2}.

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

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-x=1$$

Whakaarohia te whārite tuarua. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=2$$

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

$$y-x=1,y-3x=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-x=1$$

Whakaarohia te whārite tuarua. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=2$$

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.

$$y-x=1,y-3x=2$$

Me tango y-3x=2 mai i y-x=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-x+3x=1-2$$

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.

$$-x+3x=1-2$$

Tāpiri -x ki te 3x.

$$2x=1-2$$

Tāpiri 1 ki te -2.

$$2x=-1$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakareatia -3 ki te -\frac{1}{2}.

$$y+\frac{3}{2}=2$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

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

Kua oti te pūnaha te whakatau.

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