Whakaoti i te {l}{y=3x}{x+y=16}

$$\left\{ \begin{array} { l } { y = 3 x } \\ { x + y = 16 } \end{array} \right\}$$

Whakaoti mō y, x

$$x=4$$

$$y=12$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. 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.

$$y-3x=0,y+x=16$$

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

$$y-3x=0$$

Me tāpiri 3x ki ngā taha e rua o te whārite.

$$y=3x$$

Whakakapia te 3x mō te y ki tērā atu whārite, y+x=16.

$$3x+x=16$$

Tāpiri 3x ki te x.

$$4x=16$$

Whakawehea ngā taha e rua ki te 4.

$$x=4$$

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

$$y=3\times 4$$

Whakareatia 3 ki te 4.

$$y=12$$

Kua oti te pūnaha te whakatau.

$$y=12,x=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. 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.

$$y-3x=0,y+x=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-3\\1&1\end{matrix}\right))\left(\begin{matrix}1&-3\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&1\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&1\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&1\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-3\right)}&-\frac{-3}{1-\left(-3\right)}\\-\frac{1}{1-\left(-3\right)}&\frac{1}{1-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}0\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{3}{4}\\-\frac{1}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}0\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\times 16\\\frac{1}{4}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}12\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=12,x=4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. 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.

$$y-3x=0,y+x=16$$

Me tango y+x=16 mai i y-3x=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-3x-x=-16$$

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.

$$-3x-x=-16$$

Tāpiri -3x ki te -x.

$$-4x=-16$$

Whakawehea ngā taha e rua ki te -4.

$$x=4$$

Whakaurua te 4 mō x ki y+x=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+4=16$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$y=12$$

Kua oti te pūnaha te whakatau.

$$y=12,x=4$$