Whakaoti i te {l}{kx+9y=18}{4x-5y=20}

$$\left\{ \begin{array} { l } { kx + 9 y = 18 } \\ { 4 x - 5 y = 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{270}{5k+36}$$

$$y=-\frac{4\left(5k-18\right)}{5k+36}$$

$$k\neq -\frac{36}{5}$$

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.

$$kx+9y=18,4x-5y=20$$

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

$$kx+9y=18$$

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

$$kx=-9y+18$$

Whakawehea ngā taha e rua ki te k.

$$x=\frac{1}{k}\left(-9y+18\right)$$

Whakareatia \frac{1}{k} ki te -9y+18.

$$x=\left(-\frac{9}{k}\right)y+\frac{18}{k}$$

Whakakapia te \frac{9\left(2-y\right)}{k} mō te x ki tērā atu whārite, 4x-5y=20.

$$4\left(\left(-\frac{9}{k}\right)y+\frac{18}{k}\right)-5y=20$$

Whakareatia 4 ki te \frac{9\left(2-y\right)}{k}.

$$\left(-\frac{36}{k}\right)y+\frac{72}{k}-5y=20$$

Tāpiri -\frac{36y}{k} ki te -5y.

$$\left(-5-\frac{36}{k}\right)y+\frac{72}{k}=20$$

Me tango \frac{72}{k} mai i ngā taha e rua o te whārite.

$$\left(-5-\frac{36}{k}\right)y=20-\frac{72}{k}$$

Whakawehea ngā taha e rua ki te -\frac{36}{k}-5.

$$y=-\frac{4\left(5k-18\right)}{5k+36}$$

Whakaurua te -\frac{4\left(-18+5k\right)}{36+5k} mō y ki x=\left(-\frac{9}{k}\right)y+\frac{18}{k}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-\frac{9}{k}\right)\left(-\frac{4\left(5k-18\right)}{5k+36}\right)+\frac{18}{k}$$

Whakareatia -\frac{9}{k} ki te -\frac{4\left(-18+5k\right)}{36+5k}.

$$x=\frac{36\left(5k-18\right)}{k\left(5k+36\right)}+\frac{18}{k}$$

Tāpiri \frac{18}{k} ki te \frac{36\left(-18+5k\right)}{k\left(36+5k\right)}.

$$x=\frac{270}{5k+36}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{270}{5k+36},y=-\frac{4\left(5k-18\right)}{5k+36}$$

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.

$$kx+9y=18,4x-5y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}k&9\\4&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}k&9\\4&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}k&9\\4&-5\end{matrix}\right))\left(\begin{matrix}k&9\\4&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}k&9\\4&-5\end{matrix}\right))\left(\begin{matrix}18\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}k&9\\4&-5\end{matrix}\right))\left(\begin{matrix}18\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}k&9\\4&-5\end{matrix}\right))\left(\begin{matrix}18\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{k\left(-5\right)-9\times 4}&-\frac{9}{k\left(-5\right)-9\times 4}\\-\frac{4}{k\left(-5\right)-9\times 4}&\frac{k}{k\left(-5\right)-9\times 4}\end{matrix}\right)\left(\begin{matrix}18\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5k+36}&\frac{9}{5k+36}\\\frac{4}{5k+36}&-\frac{k}{5k+36}\end{matrix}\right)\left(\begin{matrix}18\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5k+36}\times 18+\frac{9}{5k+36}\times 20\\\frac{4}{5k+36}\times 18+\left(-\frac{k}{5k+36}\right)\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{270}{5k+36}\\\frac{4\left(18-5k\right)}{5k+36}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{270}{5k+36},y=\frac{4\left(18-5k\right)}{5k+36}$$

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.

$$kx+9y=18,4x-5y=20$$

Kia ōrite ai a kx me 4x, 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 k.

$$4kx+4\times 9y=4\times 18,k\times 4x+k\left(-5\right)y=k\times 20$$

Whakarūnātia.

$$4kx+36y=72,4kx+\left(-5k\right)y=20k$$

Me tango 4kx+\left(-5k\right)y=20k mai i 4kx+36y=72 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4kx+\left(-4k\right)x+36y+5ky=72-20k$$

Tāpiri 4kx ki te -4kx. Ka whakakore atu ngā kupu 4kx me -4kx, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$36y+5ky=72-20k$$

Tāpiri 36y ki te 5ky.

$$\left(5k+36\right)y=72-20k$$

Whakawehea ngā taha e rua ki te 36+5k.

$$y=\frac{4\left(18-5k\right)}{5k+36}$$

Whakaurua te \frac{4\left(18-5k\right)}{36+5k} mō y ki 4x-5y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-5\times \frac{4\left(18-5k\right)}{5k+36}=20$$

Whakareatia -5 ki te \frac{4\left(18-5k\right)}{36+5k}.

$$4x-\frac{20\left(18-5k\right)}{5k+36}=20$$

Me tāpiri \frac{20\left(18-5k\right)}{36+5k} ki ngā taha e rua o te whārite.

$$4x=\frac{1080}{5k+36}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{270}{5k+36}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{270}{5k+36},y=\frac{4\left(18-5k\right)}{5k+36}$$