Whakaoti i te {l}{x+y=8}{y=3x}

$$\left. \begin{array} { l } { x + y = 8 } \\ { y = 3 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-3x=0$$

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.

$$x+y=8,-3x+y=0$$

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

$$x+y=8$$

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

$$x=-y+8$$

Whakakapia te -y+8 mō te x ki tērā atu whārite, -3x+y=0.

$$-3\left(-y+8\right)+y=0$$

Whakareatia -3 ki te -y+8.

$$3y-24+y=0$$

Tāpiri 3y ki te y.

$$4y-24=0$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$4y=24$$

Whakawehea ngā taha e rua ki te 4.

$$y=6$$

Whakaurua te 6 mō y ki x=-y+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-6+8$$

Tāpiri 8 ki te -6.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=6$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-3x=0$$

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

$$x+y=8,-3x+y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\-3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\0\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\-3&1\end{matrix}\right))\left(\begin{matrix}1&1\\-3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-3&1\end{matrix}\right))\left(\begin{matrix}8\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-3&1\end{matrix}\right))\left(\begin{matrix}8\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-3&1\end{matrix}\right))\left(\begin{matrix}8\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-3\right)}&-\frac{1}{1-\left(-3\right)}\\-\frac{-3}{1-\left(-3\right)}&\frac{1}{1-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}8\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&-\frac{1}{4}\\\frac{3}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}8\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 8\\\frac{3}{4}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=6$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-3x=0$$

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.

$$x+y=8,-3x+y=0$$

Me tango -3x+y=0 mai i x+y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x+3x+y-y=8$$

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

$$x+3x=8$$

Tāpiri x ki te 3x.

$$4x=8$$

Whakawehea ngā taha e rua ki te 4.

$$x=2$$

Whakaurua te 2 mō x ki -3x+y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-3\times 2+y=0$$

Whakareatia -3 ki te 2.

$$-6+y=0$$

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

$$y=6$$

Kua oti te pūnaha te whakatau.

$$x=2,y=6$$