Whakaoti i te {l}{-3x+2y=4}{x=2-y}

$$\left. \begin{array} { l } { - 3 x + 2 y = 4 } \\ { x = 2 - y } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$x+y=2$$

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.

$$-3x+2y=4,x+y=2$$

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

$$-3x+2y=4$$

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

$$-3x=-2y+4$$

Whakawehea ngā taha e rua ki te -3.

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

Whakareatia -\frac{1}{3} ki te -2y+4.

$$x=\frac{2}{3}y-\frac{4}{3}$$

Whakakapia te \frac{-4+2y}{3} mō te x ki tērā atu whārite, x+y=2.

$$\frac{2}{3}y-\frac{4}{3}+y=2$$

Tāpiri \frac{2y}{3} ki te y.

$$\frac{5}{3}y-\frac{4}{3}=2$$

Me tāpiri \frac{4}{3} ki ngā taha e rua o te whārite.

$$\frac{5}{3}y=\frac{10}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=\frac{2}{3}y-\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\times 2-\frac{4}{3}$$

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

$$x=\frac{4-4}{3}$$

Tāpiri -\frac{4}{3} ki te \frac{4}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$x+y=2$$

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

$$-3x+2y=4,x+y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&2\\1&1\end{matrix}\right))\left(\begin{matrix}4\\2\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{2}{5}\\\frac{1}{5}&\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}4\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 4+\frac{2}{5}\times 2\\\frac{1}{5}\times 4+\frac{3}{5}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$x+y=2$$

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.

$$-3x+2y=4,x+y=2$$

Kia ōrite ai a -3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -3.

$$-3x+2y=4,-3x-3y=-3\times 2$$

Whakarūnātia.

$$-3x+2y=4,-3x-3y=-6$$

Me tango -3x-3y=-6 mai i -3x+2y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x+2y+3y=4+6$$

Tāpiri -3x ki te 3x. Ka whakakore atu ngā kupu -3x me 3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y+3y=4+6$$

Tāpiri 2y ki te 3y.

$$5y=4+6$$

Tāpiri 4 ki te 6.

$$5y=10$$

Whakawehea ngā taha e rua ki te 5.

$$y=2$$

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

$$x+2=2$$

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

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=2$$