Whakaoti i te {l}{x+2y=4}{y=2x+7}

$$\left. \begin{array} { l } { x + 2 y = 4 } \\ { y = 2 x + 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-2x=7$$

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.

$$x+2y=4,-2x+y=7$$

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

$$x+2y=4$$

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

$$x=-2y+4$$

Whakakapia te -2y+4 mō te x ki tērā atu whārite, -2x+y=7.

$$-2\left(-2y+4\right)+y=7$$

Whakareatia -2 ki te -2y+4.

$$4y-8+y=7$$

Tāpiri 4y ki te y.

$$5y-8=7$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$5y=15$$

Whakawehea ngā taha e rua ki te 5.

$$y=3$$

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

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

Whakareatia -2 ki te 3.

$$x=-6+4$$

Tāpiri 4 ki te -6.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-2x=7$$

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

$$x+2y=4,-2x+y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\-2&1\end{matrix}\right))\left(\begin{matrix}1&2\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-2&1\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-2&1\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-2&1\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-2\left(-2\right)}&-\frac{2}{1-2\left(-2\right)}\\-\frac{-2}{1-2\left(-2\right)}&\frac{1}{1-2\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}4\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&-\frac{2}{5}\\\frac{2}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}4\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 4-\frac{2}{5}\times 7\\\frac{2}{5}\times 4+\frac{1}{5}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-2x=7$$

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.

$$x+2y=4,-2x+y=7$$

Kia ōrite ai a x me -2x, 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.

$$-2x-2\times 2y=-2\times 4,-2x+y=7$$

Whakarūnātia.

$$-2x-4y=-8,-2x+y=7$$

Me tango -2x+y=7 mai i -2x-4y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-4y-y=-8-7$$

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

$$-4y-y=-8-7$$

Tāpiri -4y ki te -y.

$$-5y=-8-7$$

Tāpiri -8 ki te -7.

$$-5y=-15$$

Whakawehea ngā taha e rua ki te -5.

$$y=3$$

Whakaurua te 3 mō y ki -2x+y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+3=7$$

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

$$-2x=4$$

Whakawehea ngā taha e rua ki te -2.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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