Whakaoti i te {c}{x=2y}{5y=3x+1}

$$\left\{ \begin{array} { c } { x = 2 y } \\ { 5 y = 3 x + 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-2y=0$$

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

$$5y-3x=1$$

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.

$$x-2y=0,-3x+5y=1$$

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

$$x-2y=0$$

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

$$x=2y$$

Whakakapia te 2y mō te x ki tērā atu whārite, -3x+5y=1.

$$-3\times 2y+5y=1$$

Whakareatia -3 ki te 2y.

$$-6y+5y=1$$

Tāpiri -6y ki te 5y.

$$-y=1$$

Whakawehea ngā taha e rua ki te -1.

$$y=-1$$

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

$$x=2\left(-1\right)$$

Whakareatia 2 ki te -1.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-2y=0$$

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

$$5y-3x=1$$

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

$$x-2y=0,-3x+5y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\-3&5\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

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

$$x-2y=0$$

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

$$5y-3x=1$$

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.

$$x-2y=0,-3x+5y=1$$

Kia ōrite ai a x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-3x-3\left(-2\right)y=0,-3x+5y=1$$

Whakarūnātia.

$$-3x+6y=0,-3x+5y=1$$

Me tango -3x+5y=1 mai i -3x+6y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x+6y-5y=-1$$

Tāpiri -3x ki te 3x. Ka whakakore atu ngā kupu -3x me 3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-5y=-1$$

Tāpiri 6y ki te -5y.

$$y=-1$$

Whakaurua te -1 mō y ki -3x+5y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+5\left(-1\right)=1$$

Whakareatia 5 ki te -1.

$$-3x-5=1$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$-3x=6$$

Whakawehea ngā taha e rua ki te -3.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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