Whakaoti i te {l}{4m+9n=-35}{3m-8n=18}

$$\left\{ \begin{array} { l } { 4 m + 9 n = - 35 } \\ { 3 m - 8 n = 18 } \end{array} \right.$$

Whakaoti mō m, n

$$m=-2$$

$$n=-3$$

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.

$$4m+9n=-35,3m-8n=18$$

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

$$4m+9n=-35$$

Me tango 9n mai i ngā taha e rua o te whārite.

$$4m=-9n-35$$

Whakawehea ngā taha e rua ki te 4.

$$m=\frac{1}{4}\left(-9n-35\right)$$

Whakareatia \frac{1}{4} ki te -9n-35.

$$m=-\frac{9}{4}n-\frac{35}{4}$$

Whakakapia te \frac{-9n-35}{4} mō te m ki tērā atu whārite, 3m-8n=18.

$$3\left(-\frac{9}{4}n-\frac{35}{4}\right)-8n=18$$

Whakareatia 3 ki te \frac{-9n-35}{4}.

$$-\frac{27}{4}n-\frac{105}{4}-8n=18$$

Tāpiri -\frac{27n}{4} ki te -8n.

$$-\frac{59}{4}n-\frac{105}{4}=18$$

Me tāpiri \frac{105}{4} ki ngā taha e rua o te whārite.

$$-\frac{59}{4}n=\frac{177}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{59}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$n=-3$$

Whakaurua te -3 mō n ki m=-\frac{9}{4}n-\frac{35}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=-\frac{9}{4}\left(-3\right)-\frac{35}{4}$$

Whakareatia -\frac{9}{4} ki te -3.

$$m=\frac{27-35}{4}$$

Tāpiri -\frac{35}{4} ki te \frac{27}{4} 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.

$$m=-2$$

Kua oti te pūnaha te whakatau.

$$m=-2,n=-3$$

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.

$$4m+9n=-35,3m-8n=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&9\\3&-8\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-35\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&9\\3&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&9\\3&-8\end{matrix}\right))\left(\begin{matrix}4&9\\3&-8\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\3&-8\end{matrix}\right))\left(\begin{matrix}-35\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\3&-8\end{matrix}\right))\left(\begin{matrix}-35\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\3&-8\end{matrix}\right))\left(\begin{matrix}-35\\18\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{4\left(-8\right)-9\times 3}&-\frac{9}{4\left(-8\right)-9\times 3}\\-\frac{3}{4\left(-8\right)-9\times 3}&\frac{4}{4\left(-8\right)-9\times 3}\end{matrix}\right)\left(\begin{matrix}-35\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{8}{59}&\frac{9}{59}\\\frac{3}{59}&-\frac{4}{59}\end{matrix}\right)\left(\begin{matrix}-35\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{8}{59}\left(-35\right)+\frac{9}{59}\times 18\\\frac{3}{59}\left(-35\right)-\frac{4}{59}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-2\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=-2,n=-3$$

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.

$$4m+9n=-35,3m-8n=18$$

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

$$3\times 4m+3\times 9n=3\left(-35\right),4\times 3m+4\left(-8\right)n=4\times 18$$

Whakarūnātia.

$$12m+27n=-105,12m-32n=72$$

Me tango 12m-32n=72 mai i 12m+27n=-105 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12m-12m+27n+32n=-105-72$$

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.

$$27n+32n=-105-72$$

Tāpiri 27n ki te 32n.

$$59n=-105-72$$

Tāpiri -105 ki te -72.

$$59n=-177$$

Whakawehea ngā taha e rua ki te 59.

$$n=-3$$

Whakaurua te -3 mō n ki 3m-8n=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$3m-8\left(-3\right)=18$$

Whakareatia -8 ki te -3.

$$3m+24=18$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$3m=-6$$

Whakawehea ngā taha e rua ki te 3.

$$m=-2$$

Kua oti te pūnaha te whakatau.

$$m=-2,n=-3$$