Whakaoti i te {c}{y=2x+16}{-3x+y=20}

$$\left. \begin{array} { c } { y = 2 x + 16 } \\ { - 3 x + y = 20 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-4$$

$$y=8$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-2x=16$$

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.

$$y-2x=16,y-3x=20$$

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

$$y-2x=16$$

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

$$y=2x+16$$

Whakakapia te 16+2x mō te y ki tērā atu whārite, y-3x=20.

$$2x+16-3x=20$$

Tāpiri 2x ki te -3x.

$$-x+16=20$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$-x=4$$

Whakawehea ngā taha e rua ki te -1.

$$x=-4$$

Whakaurua te -4 mō x ki y=2x+16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2\left(-4\right)+16$$

Whakareatia 2 ki te -4.

$$y=-8+16$$

Tāpiri 16 ki te -8.

$$y=8$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-2x=16$$

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

$$y-2x=16,y-3x=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}16\\20\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right))\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right))\left(\begin{matrix}16\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right))\left(\begin{matrix}16\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&-3\end{matrix}\right))\left(\begin{matrix}16\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-\left(-2\right)}&-\frac{-2}{-3-\left(-2\right)}\\-\frac{1}{-3-\left(-2\right)}&\frac{1}{-3-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}16\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}3&-2\\1&-1\end{matrix}\right)\left(\begin{matrix}16\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}3\times 16-2\times 20\\16-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}8\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-2x=16$$

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.

$$y-2x=16,y-3x=20$$

Me tango y-3x=20 mai i y-2x=16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-2x+3x=16-20$$

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.

$$-2x+3x=16-20$$

Tāpiri -2x ki te 3x.

$$x=16-20$$

Tāpiri 16 ki te -20.

$$x=-4$$

Whakaurua te -4 mō x ki y-3x=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-3\left(-4\right)=20$$

Whakareatia -3 ki te -4.

$$y+12=20$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$y=8$$

Kua oti te pūnaha te whakatau.

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