Whakaoti i te {l}{x=2y+1}{x=3y-4}

$$\left\{ \begin{array} { l } { x = 2 y + 1 } \\ { x = 3 y - 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=11$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-2y=1$$

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x-3y=-4$$

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.

$$x-2y=1,x-3y=-4$$

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

$$x-2y=1$$

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

$$x=2y+1$$

Whakakapia te 2y+1 mō te x ki tērā atu whārite, x-3y=-4.

$$2y+1-3y=-4$$

Tāpiri 2y ki te -3y.

$$-y+1=-4$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-y=-5$$

Whakawehea ngā taha e rua ki te -1.

$$y=5$$

Whakaurua te 5 mō y ki x=2y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2\times 5+1$$

Whakareatia 2 ki te 5.

$$x=10+1$$

Tāpiri 1 ki te 10.

$$x=11$$

Kua oti te pūnaha te whakatau.

$$x=11,y=5$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-2y=1$$

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x-3y=-4$$

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

$$x-2y=1,x-3y=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-4\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right))\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right))\left(\begin{matrix}1\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right))\left(\begin{matrix}1\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right))\left(\begin{matrix}1\\-4\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-\left(-2\right)}&-\frac{-2}{-3-\left(-2\right)}\\-\frac{1}{-3-\left(-2\right)}&\frac{1}{-3-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}1\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3&-2\\1&-1\end{matrix}\right)\left(\begin{matrix}1\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3-2\left(-4\right)\\1-\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=11,y=5$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x-2y=1$$

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x-3y=-4$$

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.

$$x-2y=1,x-3y=-4$$

Me tango x-3y=-4 mai i x-2y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x-2y+3y=1+4$$

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

$$-2y+3y=1+4$$

Tāpiri -2y ki te 3y.

$$y=1+4$$

Tāpiri 1 ki te 4.

$$y=5$$

Whakaurua te 5 mō y ki x-3y=-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\times 5=-4$$

Whakareatia -3 ki te 5.

$$x-15=-4$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$x=11$$

Kua oti te pūnaha te whakatau.

$$x=11,y=5$$