Whakaoti i te {l}{3m+4n=7}{4m-3n-1=0}

$$\left\{ \begin{array} { l } { 3 m + 4 n = 7 } \\ { 4 m - 3 n - 1 = 0 } \end{array} \right.$$

Whakaoti mō m, n

$$m=1$$

$$n=1$$

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.

$$3m+4n=7,4m-3n-1=0$$

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

$$3m+4n=7$$

Me tango 4n mai i ngā taha e rua o te whārite.

$$3m=-4n+7$$

Whakawehea ngā taha e rua ki te 3.

$$m=\frac{1}{3}\left(-4n+7\right)$$

Whakareatia \frac{1}{3} ki te -4n+7.

$$m=-\frac{4}{3}n+\frac{7}{3}$$

Whakakapia te \frac{-4n+7}{3} mō te m ki tērā atu whārite, 4m-3n-1=0.

$$4\left(-\frac{4}{3}n+\frac{7}{3}\right)-3n-1=0$$

Whakareatia 4 ki te \frac{-4n+7}{3}.

$$-\frac{16}{3}n+\frac{28}{3}-3n-1=0$$

Tāpiri -\frac{16n}{3} ki te -3n.

$$-\frac{25}{3}n+\frac{28}{3}-1=0$$

Tāpiri \frac{28}{3} ki te -1.

$$-\frac{25}{3}n+\frac{25}{3}=0$$

Me tango \frac{25}{3} mai i ngā taha e rua o te whārite.

$$-\frac{25}{3}n=-\frac{25}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{25}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$n=1$$

Whakaurua te 1 mō n ki m=-\frac{4}{3}n+\frac{7}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=\frac{-4+7}{3}$$

Tāpiri \frac{7}{3} ki te -\frac{4}{3} 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.

$$m=1,n=1$$

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.

$$3m+4n=7,4m-3n-1=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\4&-3\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&4\\4&-3\end{matrix}\right))\left(\begin{matrix}3&4\\4&-3\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\4&-3\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\4&-3\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\4&-3\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{3\left(-3\right)-4\times 4}&-\frac{4}{3\left(-3\right)-4\times 4}\\-\frac{4}{3\left(-3\right)-4\times 4}&\frac{3}{3\left(-3\right)-4\times 4}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{25}&\frac{4}{25}\\\frac{4}{25}&-\frac{3}{25}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{25}\times 7+\frac{4}{25}\\\frac{4}{25}\times 7-\frac{3}{25}\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa m me n.

$$m=1,n=1$$

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.

$$3m+4n=7,4m-3n-1=0$$

Kia ōrite ai a 3m me 4m, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$4\times 3m+4\times 4n=4\times 7,3\times 4m+3\left(-3\right)n+3\left(-1\right)=0$$

Whakarūnātia.

$$12m+16n=28,12m-9n-3=0$$

Me tango 12m-9n-3=0 mai i 12m+16n=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12m-12m+16n+9n+3=28$$

Tāpiri 12m ki te -12m. Ka whakakore atu ngā kupu 12m me -12m, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$16n+9n+3=28$$

Tāpiri 16n ki te 9n.

$$25n+3=28$$

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

$$25n=25$$

Whakawehea ngā taha e rua ki te 25.

$$n=1$$

Whakaurua te 1 mō n ki 4m-3n-1=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$4m-3-1=0$$

Tāpiri -3 ki te -1.

$$4m-4=0$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$4m=4$$

Whakawehea ngā taha e rua ki te 4.

$$m=1$$

Kua oti te pūnaha te whakatau.

$$m=1,n=1$$