Whakaoti i te {l}{3w-2z=5}{w+2z=15}

$$\left. \begin{array} { l } { 3 w - 2 z = 5 } \\ { w + 2 z = 15 } \end{array} \right.$$

Whakaoti mō w, z

$$z=5$$

$$w=5$$

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.

$$3w-2z=5,w+2z=15$$

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

$$3w-2z=5$$

Me tāpiri 2z ki ngā taha e rua o te whārite.

$$3w=2z+5$$

Whakawehea ngā taha e rua ki te 3.

$$w=\frac{1}{3}\left(2z+5\right)$$

Whakareatia \frac{1}{3} ki te 2z+5.

$$w=\frac{2}{3}z+\frac{5}{3}$$

Whakakapia te \frac{2z+5}{3} mō te w ki tērā atu whārite, w+2z=15.

$$\frac{2}{3}z+\frac{5}{3}+2z=15$$

Tāpiri \frac{2z}{3} ki te 2z.

$$\frac{8}{3}z+\frac{5}{3}=15$$

Me tango \frac{5}{3} mai i ngā taha e rua o te whārite.

$$\frac{8}{3}z=\frac{40}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$z=5$$

Whakaurua te 5 mō z ki w=\frac{2}{3}z+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō w hāngai tonu.

$$w=\frac{2}{3}\times 5+\frac{5}{3}$$

Whakareatia \frac{2}{3} ki te 5.

$$w=\frac{10+5}{3}$$

Tāpiri \frac{5}{3} ki te \frac{10}{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.

$$w=5$$

Kua oti te pūnaha te whakatau.

$$w=5,z=5$$

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.

$$3w-2z=5,w+2z=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\1&2\end{matrix}\right)\left(\begin{matrix}w\\z\end{matrix}\right)=\left(\begin{matrix}5\\15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-2\\1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\1&2\end{matrix}\right))\left(\begin{matrix}3&-2\\1&2\end{matrix}\right)\left(\begin{matrix}w\\z\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\1&2\end{matrix}\right))\left(\begin{matrix}5\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}w\\z\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\1&2\end{matrix}\right))\left(\begin{matrix}5\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}w\\z\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\1&2\end{matrix}\right))\left(\begin{matrix}5\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}w\\z\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-\left(-2\right)}&-\frac{-2}{3\times 2-\left(-2\right)}\\-\frac{1}{3\times 2-\left(-2\right)}&\frac{3}{3\times 2-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}5\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}w\\z\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\-\frac{1}{8}&\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}5\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}w\\z\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 5+\frac{1}{4}\times 15\\-\frac{1}{8}\times 5+\frac{3}{8}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}w\\z\end{matrix}\right)=\left(\begin{matrix}5\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa w me z.

$$w=5,z=5$$

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.

$$3w-2z=5,w+2z=15$$

Kia ōrite ai a 3w me w, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$3w-2z=5,3w+3\times 2z=3\times 15$$

Whakarūnātia.

$$3w-2z=5,3w+6z=45$$

Me tango 3w+6z=45 mai i 3w-2z=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3w-3w-2z-6z=5-45$$

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

$$-2z-6z=5-45$$

Tāpiri -2z ki te -6z.

$$-8z=5-45$$

Tāpiri 5 ki te -45.

$$-8z=-40$$

Whakawehea ngā taha e rua ki te -8.

$$z=5$$

Whakaurua te 5 mō z ki w+2z=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō w hāngai tonu.

$$w+2\times 5=15$$

Whakareatia 2 ki te 5.

$$w+10=15$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$w=5$$

Kua oti te pūnaha te whakatau.

$$w=5,z=5$$