Whakaoti i te {l}{2m+3n=1}{7m+3n=6}

$$\left\{ \begin{array} { l } { 2 m + 3 n = 1 } \\ { 7 m + 3 n = 6 } \end{array} \right.$$

Whakaoti mō m, n

$$m=1$$

$$n=-\frac{1}{3}\approx -0.333333333$$

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.

$$2m+3n=1,7m+3n=6$$

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

$$2m+3n=1$$

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

$$2m=-3n+1$$

Whakawehea ngā taha e rua ki te 2.

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

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

$$m=-\frac{3}{2}n+\frac{1}{2}$$

Whakakapia te \frac{-3n+1}{2} mō te m ki tērā atu whārite, 7m+3n=6.

$$7\left(-\frac{3}{2}n+\frac{1}{2}\right)+3n=6$$

Whakareatia 7 ki te \frac{-3n+1}{2}.

$$-\frac{21}{2}n+\frac{7}{2}+3n=6$$

Tāpiri -\frac{21n}{2} ki te 3n.

$$-\frac{15}{2}n+\frac{7}{2}=6$$

Me tango \frac{7}{2} mai i ngā taha e rua o te whārite.

$$-\frac{15}{2}n=\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{15}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$n=-\frac{1}{3}$$

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

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

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

$$m=\frac{1+1}{2}$$

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

$$m=1$$

Kua oti te pūnaha te whakatau.

$$m=1,n=-\frac{1}{3}$$

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.

$$2m+3n=1,7m+3n=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\7&3\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}1\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\7&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\7&3\end{matrix}\right))\left(\begin{matrix}2&3\\7&3\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\7&3\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\7&3\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\7&3\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-3\times 7}&-\frac{3}{2\times 3-3\times 7}\\-\frac{7}{2\times 3-3\times 7}&\frac{2}{2\times 3-3\times 7}\end{matrix}\right)\left(\begin{matrix}1\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{1}{5}\\\frac{7}{15}&-\frac{2}{15}\end{matrix}\right)\left(\begin{matrix}1\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}+\frac{1}{5}\times 6\\\frac{7}{15}-\frac{2}{15}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa m me n.

$$m=1,n=-\frac{1}{3}$$

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.

$$2m+3n=1,7m+3n=6$$

Me tango 7m+3n=6 mai i 2m+3n=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2m-7m+3n-3n=1-6$$

Tāpiri 3n ki te -3n. Ka whakakore atu ngā kupu 3n me -3n, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2m-7m=1-6$$

Tāpiri 2m ki te -7m.

$$-5m=1-6$$

Tāpiri 1 ki te -6.

$$-5m=-5$$

Whakawehea ngā taha e rua ki te -5.

$$m=1$$

Whakaurua te 1 mō m ki 7m+3n=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō n hāngai tonu.

$$7+3n=6$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$3n=-1$$

Whakawehea ngā taha e rua ki te 3.

$$n=-\frac{1}{3}$$

Kua oti te pūnaha te whakatau.

$$m=1,n=-\frac{1}{3}$$