Whakaoti i te {l}{7w+2z=16}{5w=2z+8}

$$\left. \begin{array} { l } { 7 w + 2 z = 16 } \\ { 5 w = 2 z + 8 } \end{array} \right.$$

Whakaoti mō w, z

$$z=1$$

$$w=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 2z mai i ngā taha e rua.

$$5w-2z=8$$

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.

$$7w+2z=16,5w-2z=8$$

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

$$7w+2z=16$$

Me tango 2z mai i ngā taha e rua o te whārite.

$$7w=-2z+16$$

Whakawehea ngā taha e rua ki te 7.

$$w=\frac{1}{7}\left(-2z+16\right)$$

Whakareatia \frac{1}{7} ki te -2z+16.

$$w=-\frac{2}{7}z+\frac{16}{7}$$

Whakakapia te \frac{-2z+16}{7} mō te w ki tērā atu whārite, 5w-2z=8.

$$5\left(-\frac{2}{7}z+\frac{16}{7}\right)-2z=8$$

Whakareatia 5 ki te \frac{-2z+16}{7}.

$$-\frac{10}{7}z+\frac{80}{7}-2z=8$$

Tāpiri -\frac{10z}{7} ki te -2z.

$$-\frac{24}{7}z+\frac{80}{7}=8$$

Me tango \frac{80}{7} mai i ngā taha e rua o te whārite.

$$-\frac{24}{7}z=-\frac{24}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{24}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$z=1$$

Whakaurua te 1 mō z ki w=-\frac{2}{7}z+\frac{16}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō w hāngai tonu.

$$w=\frac{-2+16}{7}$$

Tāpiri \frac{16}{7} ki te -\frac{2}{7} 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.

$$w=2$$

Kua oti te pūnaha te whakatau.

$$w=2,z=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 2z mai i ngā taha e rua.

$$5w-2z=8$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$7w+2z=16,5w-2z=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&2\\5&-2\end{matrix}\right)\left(\begin{matrix}w\\z\end{matrix}\right)=\left(\begin{matrix}16\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&2\\5&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&2\\5&-2\end{matrix}\right))\left(\begin{matrix}7&2\\5&-2\end{matrix}\right)\left(\begin{matrix}w\\z\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\5&-2\end{matrix}\right))\left(\begin{matrix}16\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}w\\z\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\5&-2\end{matrix}\right))\left(\begin{matrix}16\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}w\\z\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\5&-2\end{matrix}\right))\left(\begin{matrix}16\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}w\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7\left(-2\right)-2\times 5}&-\frac{2}{7\left(-2\right)-2\times 5}\\-\frac{5}{7\left(-2\right)-2\times 5}&\frac{7}{7\left(-2\right)-2\times 5}\end{matrix}\right)\left(\begin{matrix}16\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}w\\z\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}&\frac{1}{12}\\\frac{5}{24}&-\frac{7}{24}\end{matrix}\right)\left(\begin{matrix}16\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}w\\z\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}\times 16+\frac{1}{12}\times 8\\\frac{5}{24}\times 16-\frac{7}{24}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}w\\z\end{matrix}\right)=\left(\begin{matrix}2\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa w me z.

$$w=2,z=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 2z mai i ngā taha e rua.

$$5w-2z=8$$

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.

$$7w+2z=16,5w-2z=8$$

Kia ōrite ai a 7w me 5w, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$5\times 7w+5\times 2z=5\times 16,7\times 5w+7\left(-2\right)z=7\times 8$$

Whakarūnātia.

$$35w+10z=80,35w-14z=56$$

Me tango 35w-14z=56 mai i 35w+10z=80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$35w-35w+10z+14z=80-56$$

Tāpiri 35w ki te -35w. Ka whakakore atu ngā kupu 35w me -35w, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10z+14z=80-56$$

Tāpiri 10z ki te 14z.

$$24z=80-56$$

Tāpiri 80 ki te -56.

$$24z=24$$

Whakawehea ngā taha e rua ki te 24.

$$z=1$$

Whakaurua te 1 mō z ki 5w-2z=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō w hāngai tonu.

$$5w-2=8$$

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

$$5w=10$$

Whakawehea ngā taha e rua ki te 5.

$$w=2$$

Kua oti te pūnaha te whakatau.

$$w=2,z=1$$