Whakaoti i te {l}{x=2y-3}{2x+5y=30}

$$\left. \begin{array}{l}{ x = 2 y - 3 }\\{ 2 x + 5 y = 30 }\end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 2y mai i ngā taha e rua.

$$x-2y=-3$$

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.

$$x-2y=-3,2x+5y=30$$

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

$$x-2y=-3$$

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

$$x=2y-3$$

Whakakapia te 2y-3 mō te x ki tērā atu whārite, 2x+5y=30.

$$2\left(2y-3\right)+5y=30$$

Whakareatia 2 ki te 2y-3.

$$4y-6+5y=30$$

Tāpiri 4y ki te 5y.

$$9y-6=30$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$9y=36$$

Whakawehea ngā taha e rua ki te 9.

$$y=4$$

Whakaurua te 4 mō y ki x=2y-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2\times 4-3$$

Whakareatia 2 ki te 4.

$$x=8-3$$

Tāpiri -3 ki te 8.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 2y mai i ngā taha e rua.

$$x-2y=-3$$

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

$$x-2y=-3,2x+5y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\2&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\2&5\end{matrix}\right))\left(\begin{matrix}1&-2\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\2&5\end{matrix}\right))\left(\begin{matrix}-3\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\2&5\end{matrix}\right))\left(\begin{matrix}-3\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\2&5\end{matrix}\right))\left(\begin{matrix}-3\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-\left(-2\times 2\right)}&-\frac{-2}{5-\left(-2\times 2\right)}\\-\frac{2}{5-\left(-2\times 2\right)}&\frac{1}{5-\left(-2\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-3\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{9}&\frac{2}{9}\\-\frac{2}{9}&\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}-3\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{9}\left(-3\right)+\frac{2}{9}\times 30\\-\frac{2}{9}\left(-3\right)+\frac{1}{9}\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 2y mai i ngā taha e rua.

$$x-2y=-3$$

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.

$$x-2y=-3,2x+5y=30$$

Kia ōrite ai a x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2x+2\left(-2\right)y=2\left(-3\right),2x+5y=30$$

Whakarūnātia.

$$2x-4y=-6,2x+5y=30$$

Me tango 2x+5y=30 mai i 2x-4y=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-4y-5y=-6-30$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y-5y=-6-30$$

Tāpiri -4y ki te -5y.

$$-9y=-6-30$$

Tāpiri -6 ki te -30.

$$-9y=-36$$

Whakawehea ngā taha e rua ki te -9.

$$y=4$$

Whakaurua te 4 mō y ki 2x+5y=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+5\times 4=30$$

Whakareatia 5 ki te 4.

$$2x+20=30$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$2x=10$$

Whakawehea ngā taha e rua ki te 2.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=4$$