Whakaoti i te {l}{m+3=8n}{m+4n=6}

$$\left. \begin{array} { l } { m + 3 = 8 n } \\ { m + 4 n = 6 } \end{array} \right.$$

Whakaoti mō m, n

$$m=3$$

$$n=\frac{3}{4}=0.75$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 8n mai i ngā taha e rua.

$$m+3-8n=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$m-8n=-3$$

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.

$$m-8n=-3,m+4n=6$$

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

$$m-8n=-3$$

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

$$m=8n-3$$

Whakakapia te 8n-3 mō te m ki tērā atu whārite, m+4n=6.

$$8n-3+4n=6$$

Tāpiri 8n ki te 4n.

$$12n-3=6$$

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

$$12n=9$$

Whakawehea ngā taha e rua ki te 12.

$$n=\frac{3}{4}$$

Whakaurua te \frac{3}{4} mō n ki m=8n-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=8\times \frac{3}{4}-3$$

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

$$m=6-3$$

Tāpiri -3 ki te 6.

$$m=3$$

Kua oti te pūnaha te whakatau.

$$m=3,n=\frac{3}{4}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 8n mai i ngā taha e rua.

$$m+3-8n=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$m-8n=-3$$

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

$$m-8n=-3,m+4n=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-8\\1&4\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-3\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-8\\1&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-8\\1&4\end{matrix}\right))\left(\begin{matrix}1&-8\\1&4\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&4\end{matrix}\right))\left(\begin{matrix}-3\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&4\end{matrix}\right))\left(\begin{matrix}-3\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&4\end{matrix}\right))\left(\begin{matrix}-3\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-\left(-8\right)}&-\frac{-8}{4-\left(-8\right)}\\-\frac{1}{4-\left(-8\right)}&\frac{1}{4-\left(-8\right)}\end{matrix}\right)\left(\begin{matrix}-3\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{2}{3}\\-\frac{1}{12}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}-3\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\left(-3\right)+\frac{2}{3}\times 6\\-\frac{1}{12}\left(-3\right)+\frac{1}{12}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}3\\\frac{3}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=3,n=\frac{3}{4}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 8n mai i ngā taha e rua.

$$m+3-8n=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$m-8n=-3$$

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.

$$m-8n=-3,m+4n=6$$

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

$$m-m-8n-4n=-3-6$$

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

$$-8n-4n=-3-6$$

Tāpiri -8n ki te -4n.

$$-12n=-3-6$$

Tāpiri -3 ki te -6.

$$-12n=-9$$

Whakawehea ngā taha e rua ki te -12.

$$n=\frac{3}{4}$$

Whakaurua te \frac{3}{4} mō n ki m+4n=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m+4\times \frac{3}{4}=6$$

Whakareatia 4 ki te \frac{3}{4}.

$$m+3=6$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$m=3$$

Kua oti te pūnaha te whakatau.

$$m=3,n=\frac{3}{4}$$