Whakaoti i te {l}{2x_{1}+3x_{2}=7}{4x_{1}-4x_{2}=-6}

$$\left. \begin{array} { l } { 2 x _ { 1 } + 3 x _ { 2 } = 7 } \\ { 4 x _ { 1 } - 4 x _ { 2 } = - 6 } \end{array} \right.$$

Whakaoti mō x_1, x_2

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

$$x_{2}=2$$

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.

$$2x_{1}+3x_{2}=7,4x_{1}-4x_{2}=-6$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x_{1} mā te wehe i te x_{1} i te taha mauī o te tohu ōrite.

$$2x_{1}+3x_{2}=7$$

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

$$2x_{1}=-3x_{2}+7$$

Whakawehea ngā taha e rua ki te 2.

$$x_{1}=\frac{1}{2}\left(-3x_{2}+7\right)$$

Whakareatia \frac{1}{2} ki te -3x_{2}+7.

$$x_{1}=-\frac{3}{2}x_{2}+\frac{7}{2}$$

Whakakapia te \frac{-3x_{2}+7}{2} mō te x_{1} ki tērā atu whārite, 4x_{1}-4x_{2}=-6.

$$4\left(-\frac{3}{2}x_{2}+\frac{7}{2}\right)-4x_{2}=-6$$

Whakareatia 4 ki te \frac{-3x_{2}+7}{2}.

$$-6x_{2}+14-4x_{2}=-6$$

Tāpiri -6x_{2} ki te -4x_{2}.

$$-10x_{2}+14=-6$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$-10x_{2}=-20$$

Whakawehea ngā taha e rua ki te -10.

$$x_{2}=2$$

Whakaurua te 2 mō x_{2} ki x_{1}=-\frac{3}{2}x_{2}+\frac{7}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x_{1} hāngai tonu.

$$x_{1}=-\frac{3}{2}\times 2+\frac{7}{2}$$

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

$$x_{1}=-3+\frac{7}{2}$$

Tāpiri \frac{7}{2} ki te -3.

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

Kua oti te pūnaha te whakatau.

$$x_{1}=\frac{1}{2},x_{2}=2$$

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.

$$2x_{1}+3x_{2}=7,4x_{1}-4x_{2}=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\4&-4\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}7\\-6\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\4&-4\end{matrix}\right))\left(\begin{matrix}2&3\\4&-4\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&-4\end{matrix}\right))\left(\begin{matrix}7\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&-4\end{matrix}\right))\left(\begin{matrix}7\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&-4\end{matrix}\right))\left(\begin{matrix}7\\-6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{2\left(-4\right)-3\times 4}&-\frac{3}{2\left(-4\right)-3\times 4}\\-\frac{4}{2\left(-4\right)-3\times 4}&\frac{2}{2\left(-4\right)-3\times 4}\end{matrix}\right)\left(\begin{matrix}7\\-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{3}{20}\\\frac{1}{5}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}7\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 7+\frac{3}{20}\left(-6\right)\\\frac{1}{5}\times 7-\frac{1}{10}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x_{1} me x_{2}.

$$x_{1}=\frac{1}{2},x_{2}=2$$

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.

$$2x_{1}+3x_{2}=7,4x_{1}-4x_{2}=-6$$

Kia ōrite ai a 2x_{1} me 4x_{1}, 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 2.

$$4\times 2x_{1}+4\times 3x_{2}=4\times 7,2\times 4x_{1}+2\left(-4\right)x_{2}=2\left(-6\right)$$

Whakarūnātia.

$$8x_{1}+12x_{2}=28,8x_{1}-8x_{2}=-12$$

Me tango 8x_{1}-8x_{2}=-12 mai i 8x_{1}+12x_{2}=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x_{1}-8x_{1}+12x_{2}+8x_{2}=28+12$$

Tāpiri 8x_{1} ki te -8x_{1}. Ka whakakore atu ngā kupu 8x_{1} me -8x_{1}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12x_{2}+8x_{2}=28+12$$

Tāpiri 12x_{2} ki te 8x_{2}.

$$20x_{2}=28+12$$

Tāpiri 28 ki te 12.

$$20x_{2}=40$$

Whakawehea ngā taha e rua ki te 20.

$$x_{2}=2$$

Whakaurua te 2 mō x_{2} ki 4x_{1}-4x_{2}=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x_{1} hāngai tonu.

$$4x_{1}-4\times 2=-6$$

Whakareatia -4 ki te 2.

$$4x_{1}-8=-6$$

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

$$4x_{1}=2$$

Whakawehea ngā taha e rua ki te 4.

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

Kua oti te pūnaha te whakatau.

$$x_{1}=\frac{1}{2},x_{2}=2$$