Whakaoti i te {l}{x+2y=7}{4x+3y=3}

$$\left\{ \begin{array} { l } { x + 2 y = 7 } \\ { 4 x + 3 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$x+2y=7,4x+3y=3$$

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

$$x+2y=7$$

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

$$x=-2y+7$$

Whakakapia te -2y+7 mō te x ki tērā atu whārite, 4x+3y=3.

$$4\left(-2y+7\right)+3y=3$$

Whakareatia 4 ki te -2y+7.

$$-8y+28+3y=3$$

Tāpiri -8y ki te 3y.

$$-5y+28=3$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$-5y=-25$$

Whakawehea ngā taha e rua ki te -5.

$$y=5$$

Whakaurua te 5 mō y ki x=-2y+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times 5+7$$

Whakareatia -2 ki te 5.

$$x=-10+7$$

Tāpiri 7 ki te -10.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=5$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x+2y=7,4x+3y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\4&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\4&3\end{matrix}\right))\left(\begin{matrix}1&2\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&3\end{matrix}\right))\left(\begin{matrix}7\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&3\end{matrix}\right))\left(\begin{matrix}7\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&3\end{matrix}\right))\left(\begin{matrix}7\\3\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-2\times 4}&-\frac{2}{3-2\times 4}\\-\frac{4}{3-2\times 4}&\frac{1}{3-2\times 4}\end{matrix}\right)\left(\begin{matrix}7\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5}&\frac{2}{5}\\\frac{4}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}7\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5}\times 7+\frac{2}{5}\times 3\\\frac{4}{5}\times 7-\frac{1}{5}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=5$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$x+2y=7,4x+3y=3$$

Kia ōrite ai a x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$4x+4\times 2y=4\times 7,4x+3y=3$$

Whakarūnātia.

$$4x+8y=28,4x+3y=3$$

Me tango 4x+3y=3 mai i 4x+8y=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+8y-3y=28-3$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-3y=28-3$$

Tāpiri 8y ki te -3y.

$$5y=28-3$$

Tāpiri 28 ki te -3.

$$5y=25$$

Whakawehea ngā taha e rua ki te 5.

$$y=5$$

Whakaurua te 5 mō y ki 4x+3y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+3\times 5=3$$

Whakareatia 3 ki te 5.

$$4x+15=3$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$4x=-12$$

Whakawehea ngā taha e rua ki te 4.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=5$$