Whakaoti i te {l}{2m-3n=130}{-m+5=4n}

$$\left\{ \begin{array} { l } { 2 m - 3 n = 130 } \\ { - m + 5 = 4 n } \end{array} \right.$$

Whakaoti mō m, n

$$m = \frac{535}{11} = 48\frac{7}{11} \approx 48.636363636$$

$$n = -\frac{120}{11} = -10\frac{10}{11} \approx -10.909090909$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 4n mai i ngā taha e rua.

$$-m+5-4n=0$$

Tangohia te 5 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-m-4n=-5$$

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.

$$2m-3n=130,-m-4n=-5$$

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

$$2m-3n=130$$

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

$$2m=3n+130$$

Whakawehea ngā taha e rua ki te 2.

$$m=\frac{1}{2}\left(3n+130\right)$$

Whakareatia \frac{1}{2} ki te 3n+130.

$$m=\frac{3}{2}n+65$$

Whakakapia te \frac{3n}{2}+65 mō te m ki tērā atu whārite, -m-4n=-5.

$$-\left(\frac{3}{2}n+65\right)-4n=-5$$

Whakareatia -1 ki te \frac{3n}{2}+65.

$$-\frac{3}{2}n-65-4n=-5$$

Tāpiri -\frac{3n}{2} ki te -4n.

$$-\frac{11}{2}n-65=-5$$

Me tāpiri 65 ki ngā taha e rua o te whārite.

$$-\frac{11}{2}n=60$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$n=-\frac{120}{11}$$

Whakaurua te -\frac{120}{11} mō n ki m=\frac{3}{2}n+65. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=\frac{3}{2}\left(-\frac{120}{11}\right)+65$$

Whakareatia \frac{3}{2} ki te -\frac{120}{11} 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.

$$m=-\frac{180}{11}+65$$

Tāpiri 65 ki te -\frac{180}{11}.

$$m=\frac{535}{11}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{535}{11},n=-\frac{120}{11}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 4n mai i ngā taha e rua.

$$-m+5-4n=0$$

Tangohia te 5 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-m-4n=-5$$

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

$$2m-3n=130,-m-4n=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\-1&-4\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}130\\-5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-3\\-1&-4\end{matrix}\right))\left(\begin{matrix}2&-3\\-1&-4\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\-1&-4\end{matrix}\right))\left(\begin{matrix}130\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\-1&-4\end{matrix}\right))\left(\begin{matrix}130\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\-1&-4\end{matrix}\right))\left(\begin{matrix}130\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{2\left(-4\right)-\left(-3\left(-1\right)\right)}&-\frac{-3}{2\left(-4\right)-\left(-3\left(-1\right)\right)}\\-\frac{-1}{2\left(-4\right)-\left(-3\left(-1\right)\right)}&\frac{2}{2\left(-4\right)-\left(-3\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}130\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}&-\frac{3}{11}\\-\frac{1}{11}&-\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}130\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}\times 130-\frac{3}{11}\left(-5\right)\\-\frac{1}{11}\times 130-\frac{2}{11}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{535}{11}\\-\frac{120}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=\frac{535}{11},n=-\frac{120}{11}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 4n mai i ngā taha e rua.

$$-m+5-4n=0$$

Tangohia te 5 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-m-4n=-5$$

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.

$$2m-3n=130,-m-4n=-5$$

Kia ōrite ai a 2m me -m, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-2m-\left(-3n\right)=-130,2\left(-1\right)m+2\left(-4\right)n=2\left(-5\right)$$

Whakarūnātia.

$$-2m+3n=-130,-2m-8n=-10$$

Me tango -2m-8n=-10 mai i -2m+3n=-130 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2m+2m+3n+8n=-130+10$$

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

$$3n+8n=-130+10$$

Tāpiri 3n ki te 8n.

$$11n=-130+10$$

Tāpiri -130 ki te 10.

$$11n=-120$$

Whakawehea ngā taha e rua ki te 11.

$$n=-\frac{120}{11}$$

Whakaurua te -\frac{120}{11} mō n ki -m-4n=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$-m-4\left(-\frac{120}{11}\right)=-5$$

Whakareatia -4 ki te -\frac{120}{11}.

$$-m+\frac{480}{11}=-5$$

Me tango \frac{480}{11} mai i ngā taha e rua o te whārite.

$$-m=-\frac{535}{11}$$

Whakawehea ngā taha e rua ki te -1.

$$m=\frac{535}{11}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{535}{11},n=-\frac{120}{11}$$