Whakaoti i te {l}{y=-3x+5}{y=2x}

$$\left. \begin{array} { l } { y = - 3 x + 5 } \\ { y = 2 x } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=5$$

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

$$y-2x=0$$

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.

$$y+3x=5,y-2x=0$$

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

$$y+3x=5$$

Me tango 3x mai i ngā taha e rua o te whārite.

$$y=-3x+5$$

Whakakapia te -3x+5 mō te y ki tērā atu whārite, y-2x=0.

$$-3x+5-2x=0$$

Tāpiri -3x ki te -2x.

$$-5x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-5x=-5$$

Whakawehea ngā taha e rua ki te -5.

$$x=1$$

Whakaurua te 1 mō x ki y=-3x+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-3+5$$

Tāpiri 5 ki te -3.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$y=2,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=5$$

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

$$y-2x=0$$

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

$$y+3x=5,y-2x=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\1&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}5\\0\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&3\\1&-2\end{matrix}\right))\left(\begin{matrix}1&3\\1&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\1&-2\end{matrix}\right))\left(\begin{matrix}5\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\1&-2\end{matrix}\right))\left(\begin{matrix}5\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\1&-2\end{matrix}\right))\left(\begin{matrix}5\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-3}&-\frac{3}{-2-3}\\-\frac{1}{-2-3}&\frac{1}{-2-3}\end{matrix}\right)\left(\begin{matrix}5\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&\frac{3}{5}\\\frac{1}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}5\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 5\\\frac{1}{5}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=2,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=5$$

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

$$y-2x=0$$

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.

$$y+3x=5,y-2x=0$$

Me tango y-2x=0 mai i y+3x=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+3x+2x=5$$

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.

$$3x+2x=5$$

Tāpiri 3x ki te 2x.

$$5x=5$$

Whakawehea ngā taha e rua ki te 5.

$$x=1$$

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

$$y-2=0$$

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

$$y=2$$

Kua oti te pūnaha te whakatau.

$$y=2,x=1$$