Whakaoti i te {l}{5x=7z}{8x=9z}

$$\left. \begin{array} { l } { 5 x = 7 z } \\ { 8 x = 9 z } \end{array} \right.$$

Whakaoti mō x, z

$$x=0$$

$$z=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 7z mai i ngā taha e rua.

$$5x-7z=0$$

Whakaarohia te whārite tuarua. Tangohia te 9z mai i ngā taha e rua.

$$8x-9z=0$$

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.

$$5x-7z=0,8x-9z=0$$

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

$$5x-7z=0$$

Me tāpiri 7z ki ngā taha e rua o te whārite.

$$5x=7z$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\times 7z$$

Whakareatia \frac{1}{5} ki te 7z.

$$x=\frac{7}{5}z$$

Whakakapia te \frac{7z}{5} mō te x ki tērā atu whārite, 8x-9z=0.

$$8\times \frac{7}{5}z-9z=0$$

Whakareatia 8 ki te \frac{7z}{5}.

$$\frac{56}{5}z-9z=0$$

Tāpiri \frac{56z}{5} ki te -9z.

$$\frac{11}{5}z=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$z=0$$

Whakaurua te 0 mō z ki x=\frac{7}{5}z. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,z=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 7z mai i ngā taha e rua.

$$5x-7z=0$$

Whakaarohia te whārite tuarua. Tangohia te 9z mai i ngā taha e rua.

$$8x-9z=0$$

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

$$5x-7z=0,8x-9z=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-7\\8&-9\end{matrix}\right)\left(\begin{matrix}x\\z\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-7\\8&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-7\\8&-9\end{matrix}\right))\left(\begin{matrix}5&-7\\8&-9\end{matrix}\right)\left(\begin{matrix}x\\z\end{matrix}\right)=inverse(\left(\begin{matrix}5&-7\\8&-9\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\z\end{matrix}\right)=inverse(\left(\begin{matrix}5&-7\\8&-9\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\z\end{matrix}\right)=inverse(\left(\begin{matrix}5&-7\\8&-9\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{5\left(-9\right)-\left(-7\times 8\right)}&-\frac{-7}{5\left(-9\right)-\left(-7\times 8\right)}\\-\frac{8}{5\left(-9\right)-\left(-7\times 8\right)}&\frac{5}{5\left(-9\right)-\left(-7\times 8\right)}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{11}&\frac{7}{11}\\-\frac{8}{11}&\frac{5}{11}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\z\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me z.

$$x=0,z=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 7z mai i ngā taha e rua.

$$5x-7z=0$$

Whakaarohia te whārite tuarua. Tangohia te 9z mai i ngā taha e rua.

$$8x-9z=0$$

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.

$$5x-7z=0,8x-9z=0$$

Kia ōrite ai a 5x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$8\times 5x+8\left(-7\right)z=0,5\times 8x+5\left(-9\right)z=0$$

Whakarūnātia.

$$40x-56z=0,40x-45z=0$$

Me tango 40x-45z=0 mai i 40x-56z=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$40x-40x-56z+45z=0$$

Tāpiri 40x ki te -40x. Ka whakakore atu ngā kupu 40x me -40x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-56z+45z=0$$

Tāpiri -56z ki te 45z.

$$-11z=0$$

Whakawehea ngā taha e rua ki te -11.

$$z=0$$

Whakaurua te 0 mō z ki 8x-9z=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x=0$$

Whakawehea ngā taha e rua ki te 8.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,z=0$$