Whakaoti i te {l}{1=2r-3s}{4=3r+2s}

$$\left. \begin{array} { l } { 1 = 2 r - 3 s } \\ { 4 = 3 r + 2 s } \end{array} \right.$$

Whakaoti mō r, s

$$r = \frac{14}{13} = 1\frac{1}{13} \approx 1.076923077$$

$$s=\frac{5}{13}\approx 0.384615385$$

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.

$$2r-3s=1$$

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

$$3r+2s=4$$

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.

$$2r-3s=1,3r+2s=4$$

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

$$2r-3s=1$$

Me tāpiri 3s ki ngā taha e rua o te whārite.

$$2r=3s+1$$

Whakawehea ngā taha e rua ki te 2.

$$r=\frac{1}{2}\left(3s+1\right)$$

Whakareatia \frac{1}{2} ki te 3s+1.

$$r=\frac{3}{2}s+\frac{1}{2}$$

Whakakapia te \frac{3s+1}{2} mō te r ki tērā atu whārite, 3r+2s=4.

$$3\left(\frac{3}{2}s+\frac{1}{2}\right)+2s=4$$

Whakareatia 3 ki te \frac{3s+1}{2}.

$$\frac{9}{2}s+\frac{3}{2}+2s=4$$

Tāpiri \frac{9s}{2} ki te 2s.

$$\frac{13}{2}s+\frac{3}{2}=4$$

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

$$\frac{13}{2}s=\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$s=\frac{5}{13}$$

Whakaurua te \frac{5}{13} mō s ki r=\frac{3}{2}s+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō r hāngai tonu.

$$r=\frac{3}{2}\times \frac{5}{13}+\frac{1}{2}$$

Whakareatia \frac{3}{2} ki te \frac{5}{13} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$r=\frac{15}{26}+\frac{1}{2}$$

Tāpiri \frac{1}{2} ki te \frac{15}{26} 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.

$$r=\frac{14}{13}$$

Kua oti te pūnaha te whakatau.

$$r=\frac{14}{13},s=\frac{5}{13}$$

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.

$$2r-3s=1$$

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

$$3r+2s=4$$

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

$$2r-3s=1,3r+2s=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}r\\s\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}&\frac{3}{13}\\-\frac{3}{13}&\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}1\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}r\\s\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}+\frac{3}{13}\times 4\\-\frac{3}{13}+\frac{2}{13}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}r\\s\end{matrix}\right)=\left(\begin{matrix}\frac{14}{13}\\\frac{5}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa r me s.

$$r=\frac{14}{13},s=\frac{5}{13}$$

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.

$$2r-3s=1$$

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

$$3r+2s=4$$

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.

$$2r-3s=1,3r+2s=4$$

Kia ōrite ai a 2r me 3r, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$3\times 2r+3\left(-3\right)s=3,2\times 3r+2\times 2s=2\times 4$$

Whakarūnātia.

$$6r-9s=3,6r+4s=8$$

Me tango 6r+4s=8 mai i 6r-9s=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6r-6r-9s-4s=3-8$$

Tāpiri 6r ki te -6r. Ka whakakore atu ngā kupu 6r me -6r, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9s-4s=3-8$$

Tāpiri -9s ki te -4s.

$$-13s=3-8$$

Tāpiri 3 ki te -8.

$$-13s=-5$$

Whakawehea ngā taha e rua ki te -13.

$$s=\frac{5}{13}$$

Whakaurua te \frac{5}{13} mō s ki 3r+2s=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō r hāngai tonu.

$$3r+2\times \frac{5}{13}=4$$

Whakareatia 2 ki te \frac{5}{13}.

$$3r+\frac{10}{13}=4$$

Me tango \frac{10}{13} mai i ngā taha e rua o te whārite.

$$3r=\frac{42}{13}$$

Whakawehea ngā taha e rua ki te 3.

$$r=\frac{14}{13}$$

Kua oti te pūnaha te whakatau.

$$r=\frac{14}{13},s=\frac{5}{13}$$