Whakaoti i te {l}{4=3A-9D}{2=8A-8D}

$$\left. \begin{array} { l } { 4 = 3 A - 9 D } \\ { 2 = 8 A - 8 D } \end{array} \right.$$

Whakaoti mō A, D

$$A=-\frac{7}{24}\approx -0.291666667$$

$$D=-\frac{13}{24}\approx -0.541666667$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3A-9D=4$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$8A-8D=2$$

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.

$$3A-9D=4,8A-8D=2$$

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

$$3A-9D=4$$

Me tāpiri 9D ki ngā taha e rua o te whārite.

$$3A=9D+4$$

Whakawehea ngā taha e rua ki te 3.

$$A=\frac{1}{3}\left(9D+4\right)$$

Whakareatia \frac{1}{3} ki te 9D+4.

$$A=3D+\frac{4}{3}$$

Whakakapia te 3D+\frac{4}{3} mō te A ki tērā atu whārite, 8A-8D=2.

$$8\left(3D+\frac{4}{3}\right)-8D=2$$

Whakareatia 8 ki te 3D+\frac{4}{3}.

$$24D+\frac{32}{3}-8D=2$$

Tāpiri 24D ki te -8D.

$$16D+\frac{32}{3}=2$$

Me tango \frac{32}{3} mai i ngā taha e rua o te whārite.

$$16D=-\frac{26}{3}$$

Whakawehea ngā taha e rua ki te 16.

$$D=-\frac{13}{24}$$

Whakaurua te -\frac{13}{24} mō D ki A=3D+\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō A hāngai tonu.

$$A=3\left(-\frac{13}{24}\right)+\frac{4}{3}$$

Whakareatia 3 ki te -\frac{13}{24}.

$$A=-\frac{13}{8}+\frac{4}{3}$$

Tāpiri \frac{4}{3} ki te -\frac{13}{8} 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.

$$A=-\frac{7}{24}$$

Kua oti te pūnaha te whakatau.

$$A=-\frac{7}{24},D=-\frac{13}{24}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3A-9D=4$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$8A-8D=2$$

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

$$3A-9D=4,8A-8D=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-9\\8&-8\end{matrix}\right)\left(\begin{matrix}A\\D\end{matrix}\right)=\left(\begin{matrix}4\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-9\\8&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-9\\8&-8\end{matrix}\right))\left(\begin{matrix}3&-9\\8&-8\end{matrix}\right)\left(\begin{matrix}A\\D\end{matrix}\right)=inverse(\left(\begin{matrix}3&-9\\8&-8\end{matrix}\right))\left(\begin{matrix}4\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}A\\D\end{matrix}\right)=inverse(\left(\begin{matrix}3&-9\\8&-8\end{matrix}\right))\left(\begin{matrix}4\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}A\\D\end{matrix}\right)=inverse(\left(\begin{matrix}3&-9\\8&-8\end{matrix}\right))\left(\begin{matrix}4\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}A\\D\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{3\left(-8\right)-\left(-9\times 8\right)}&-\frac{-9}{3\left(-8\right)-\left(-9\times 8\right)}\\-\frac{8}{3\left(-8\right)-\left(-9\times 8\right)}&\frac{3}{3\left(-8\right)-\left(-9\times 8\right)}\end{matrix}\right)\left(\begin{matrix}4\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}A\\D\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}&\frac{3}{16}\\-\frac{1}{6}&\frac{1}{16}\end{matrix}\right)\left(\begin{matrix}4\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}A\\D\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}\times 4+\frac{3}{16}\times 2\\-\frac{1}{6}\times 4+\frac{1}{16}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}A\\D\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{24}\\-\frac{13}{24}\end{matrix}\right)$$

Tangohia ngā huānga poukapa A me D.

$$A=-\frac{7}{24},D=-\frac{13}{24}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3A-9D=4$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$8A-8D=2$$

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.

$$3A-9D=4,8A-8D=2$$

Kia ōrite ai a 3A me 8A, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$8\times 3A+8\left(-9\right)D=8\times 4,3\times 8A+3\left(-8\right)D=3\times 2$$

Whakarūnātia.

$$24A-72D=32,24A-24D=6$$

Me tango 24A-24D=6 mai i 24A-72D=32 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24A-24A-72D+24D=32-6$$

Tāpiri 24A ki te -24A. Ka whakakore atu ngā kupu 24A me -24A, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-72D+24D=32-6$$

Tāpiri -72D ki te 24D.

$$-48D=32-6$$

Tāpiri 32 ki te -6.

$$-48D=26$$

Whakawehea ngā taha e rua ki te -48.

$$D=-\frac{13}{24}$$

Whakaurua te -\frac{13}{24} mō D ki 8A-8D=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō A hāngai tonu.

$$8A-8\left(-\frac{13}{24}\right)=2$$

Whakareatia -8 ki te -\frac{13}{24}.

$$8A+\frac{13}{3}=2$$

Me tango \frac{13}{3} mai i ngā taha e rua o te whārite.

$$8A=-\frac{7}{3}$$

Whakawehea ngā taha e rua ki te 8.

$$A=-\frac{7}{24}$$

Kua oti te pūnaha te whakatau.

$$A=-\frac{7}{24},D=-\frac{13}{24}$$