Whakaoti i te {l}{16m+50n=55}{2m+4n=5}

$$\left\{ \begin{array} { l } { 16 m + 50 n = 55 } \\ { 2 m + 4 n = 5 } \end{array} \right.$$

Whakaoti mō m, n

$$m=\frac{5}{6}\approx 0.833333333$$

$$n=\frac{5}{6}\approx 0.833333333$$

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.

$$16m+50n=55,2m+4n=5$$

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

$$16m+50n=55$$

Me tango 50n mai i ngā taha e rua o te whārite.

$$16m=-50n+55$$

Whakawehea ngā taha e rua ki te 16.

$$m=\frac{1}{16}\left(-50n+55\right)$$

Whakareatia \frac{1}{16} ki te -50n+55.

$$m=-\frac{25}{8}n+\frac{55}{16}$$

Whakakapia te -\frac{25n}{8}+\frac{55}{16} mō te m ki tērā atu whārite, 2m+4n=5.

$$2\left(-\frac{25}{8}n+\frac{55}{16}\right)+4n=5$$

Whakareatia 2 ki te -\frac{25n}{8}+\frac{55}{16}.

$$-\frac{25}{4}n+\frac{55}{8}+4n=5$$

Tāpiri -\frac{25n}{4} ki te 4n.

$$-\frac{9}{4}n+\frac{55}{8}=5$$

Me tango \frac{55}{8} mai i ngā taha e rua o te whārite.

$$-\frac{9}{4}n=-\frac{15}{8}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$n=\frac{5}{6}$$

Whakaurua te \frac{5}{6} mō n ki m=-\frac{25}{8}n+\frac{55}{16}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=-\frac{25}{8}\times \frac{5}{6}+\frac{55}{16}$$

Whakareatia -\frac{25}{8} ki te \frac{5}{6} 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.

$$m=-\frac{125}{48}+\frac{55}{16}$$

Tāpiri \frac{55}{16} ki te -\frac{125}{48} 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.

$$m=\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{5}{6},n=\frac{5}{6}$$

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.

$$16m+50n=55,2m+4n=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}16&50\\2&4\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}55\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}16&50\\2&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}16&50\\2&4\end{matrix}\right))\left(\begin{matrix}16&50\\2&4\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}16&50\\2&4\end{matrix}\right))\left(\begin{matrix}55\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}16&50\\2&4\end{matrix}\right))\left(\begin{matrix}55\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}16&50\\2&4\end{matrix}\right))\left(\begin{matrix}55\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{4}{16\times 4-50\times 2}&-\frac{50}{16\times 4-50\times 2}\\-\frac{2}{16\times 4-50\times 2}&\frac{16}{16\times 4-50\times 2}\end{matrix}\right)\left(\begin{matrix}55\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{9}&\frac{25}{18}\\\frac{1}{18}&-\frac{4}{9}\end{matrix}\right)\left(\begin{matrix}55\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{9}\times 55+\frac{25}{18}\times 5\\\frac{1}{18}\times 55-\frac{4}{9}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}\\\frac{5}{6}\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=\frac{5}{6},n=\frac{5}{6}$$

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.

$$16m+50n=55,2m+4n=5$$

Kia ōrite ai a 16m me 2m, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 16.

$$2\times 16m+2\times 50n=2\times 55,16\times 2m+16\times 4n=16\times 5$$

Whakarūnātia.

$$32m+100n=110,32m+64n=80$$

Me tango 32m+64n=80 mai i 32m+100n=110 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$32m-32m+100n-64n=110-80$$

Tāpiri 32m ki te -32m. Ka whakakore atu ngā kupu 32m me -32m, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$100n-64n=110-80$$

Tāpiri 100n ki te -64n.

$$36n=110-80$$

Tāpiri 110 ki te -80.

$$36n=30$$

Whakawehea ngā taha e rua ki te 36.

$$n=\frac{5}{6}$$

Whakaurua te \frac{5}{6} mō n ki 2m+4n=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$2m+4\times \frac{5}{6}=5$$

Whakareatia 4 ki te \frac{5}{6}.

$$2m+\frac{10}{3}=5$$

Me tango \frac{10}{3} mai i ngā taha e rua o te whārite.

$$2m=\frac{5}{3}$$

Whakawehea ngā taha e rua ki te 2.

$$m=\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{5}{6},n=\frac{5}{6}$$