Whakaoti i te {l}{x+2y=1}{y=4x-5}

$$\left. \begin{array} { l } { x + 2 y = 1 } \\ { y = 4 x - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{11}{9} = 1\frac{2}{9} \approx 1.222222222$$

$$y=-\frac{1}{9}\approx -0.111111111$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=-5$$

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.

$$x+2y=1,-4x+y=-5$$

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 tango 2y mai i ngā taha e rua o te whārite.

$$x=-2y+1$$

Whakakapia te -2y+1 mō te x ki tērā atu whārite, -4x+y=-5.

$$-4\left(-2y+1\right)+y=-5$$

Whakareatia -4 ki te -2y+1.

$$8y-4+y=-5$$

Tāpiri 8y ki te y.

$$9y-4=-5$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$9y=-1$$

Whakawehea ngā taha e rua ki te 9.

$$y=-\frac{1}{9}$$

Whakaurua te -\frac{1}{9} 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.

$$x=-2\left(-\frac{1}{9}\right)+1$$

Whakareatia -2 ki te -\frac{1}{9}.

$$x=\frac{2}{9}+1$$

Tāpiri 1 ki te \frac{2}{9}.

$$x=\frac{11}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{11}{9},y=-\frac{1}{9}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=-5$$

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

$$x+2y=1,-4x+y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\-4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\-4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\-4&1\end{matrix}\right))\left(\begin{matrix}1&2\\-4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-4&1\end{matrix}\right))\left(\begin{matrix}1\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-4&1\end{matrix}\right))\left(\begin{matrix}1\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-4&1\end{matrix}\right))\left(\begin{matrix}1\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-2\left(-4\right)}&-\frac{2}{1-2\left(-4\right)}\\-\frac{-4}{1-2\left(-4\right)}&\frac{1}{1-2\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}1\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&-\frac{2}{9}\\\frac{4}{9}&\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}1\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}-\frac{2}{9}\left(-5\right)\\\frac{4}{9}+\frac{1}{9}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{9}\\-\frac{1}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{11}{9},y=-\frac{1}{9}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=-5$$

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.

$$x+2y=1,-4x+y=-5$$

Kia ōrite ai a x 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 1.

$$-4x-4\times 2y=-4,-4x+y=-5$$

Whakarūnātia.

$$-4x-8y=-4,-4x+y=-5$$

Me tango -4x+y=-5 mai i -4x-8y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x-8y-y=-4+5$$

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

$$-8y-y=-4+5$$

Tāpiri -8y ki te -y.

$$-9y=-4+5$$

Tāpiri -4 ki te 5.

$$-9y=1$$

Whakawehea ngā taha e rua ki te -9.

$$y=-\frac{1}{9}$$

Whakaurua te -\frac{1}{9} mō y ki -4x+y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x-\frac{1}{9}=-5$$

Me tāpiri \frac{1}{9} ki ngā taha e rua o te whārite.

$$-4x=-\frac{44}{9}$$

Whakawehea ngā taha e rua ki te -4.

$$x=\frac{11}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{11}{9},y=-\frac{1}{9}$$