Whakaoti i te {l}{y=2x+7}{2y=x+2}

$$\left. \begin{array} { l } { y = 2 x + 7 } \\ { 2 y = x + 2 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-4$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-2x=7$$

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

$$2y-x=2$$

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.

$$y-2x=7,2y-x=2$$

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

$$y-2x=7$$

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

$$y=2x+7$$

Whakakapia te 2x+7 mō te y ki tērā atu whārite, 2y-x=2.

$$2\left(2x+7\right)-x=2$$

Whakareatia 2 ki te 2x+7.

$$4x+14-x=2$$

Tāpiri 4x ki te -x.

$$3x+14=2$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$3x=-12$$

Whakawehea ngā taha e rua ki te 3.

$$x=-4$$

Whakaurua te -4 mō x ki y=2x+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2\left(-4\right)+7$$

Whakareatia 2 ki te -4.

$$y=-8+7$$

Tāpiri 7 ki te -8.

$$y=-1$$

Kua oti te pūnaha te whakatau.

$$y=-1,x=-4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-2x=7$$

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

$$2y-x=2$$

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

$$y-2x=7,2y-x=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\2&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7\\2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-2\\2&-1\end{matrix}\right))\left(\begin{matrix}1&-2\\2&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\2&-1\end{matrix}\right))\left(\begin{matrix}7\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\2&-1\end{matrix}\right))\left(\begin{matrix}7\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\2&-1\end{matrix}\right))\left(\begin{matrix}7\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-2\times 2\right)}&-\frac{-2}{-1-\left(-2\times 2\right)}\\-\frac{2}{-1-\left(-2\times 2\right)}&\frac{1}{-1-\left(-2\times 2\right)}\end{matrix}\right)\left(\begin{matrix}7\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{2}{3}\\-\frac{2}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}7\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 7+\frac{2}{3}\times 2\\-\frac{2}{3}\times 7+\frac{1}{3}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=-1,x=-4$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-2x=7$$

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

$$2y-x=2$$

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.

$$y-2x=7,2y-x=2$$

Kia ōrite ai a y me 2y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2y+2\left(-2\right)x=2\times 7,2y-x=2$$

Whakarūnātia.

$$2y-4x=14,2y-x=2$$

Me tango 2y-x=2 mai i 2y-4x=14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2y-2y-4x+x=14-2$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4x+x=14-2$$

Tāpiri -4x ki te x.

$$-3x=14-2$$

Tāpiri 14 ki te -2.

$$-3x=12$$

Whakawehea ngā taha e rua ki te -3.

$$x=-4$$

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

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

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

$$2y=-2$$

Whakawehea ngā taha e rua ki te 2.

$$y=-1$$

Kua oti te pūnaha te whakatau.

$$y=-1,x=-4$$