Whakaoti i te {l}{y=2x+1}{5y-7x=11}

$$\left\{ \begin{array} { l } { y = 2 x + 1 } \\ { 5 y - 7 x = 11 } \end{array} \right.$$

Whakaoti mō y, x

$$x=2$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-2x=1$$

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.

$$y-2x=1,5y-7x=11$$

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

$$y-2x=1$$

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

$$y=2x+1$$

Whakakapia te 2x+1 mō te y ki tērā atu whārite, 5y-7x=11.

$$5\left(2x+1\right)-7x=11$$

Whakareatia 5 ki te 2x+1.

$$10x+5-7x=11$$

Tāpiri 10x ki te -7x.

$$3x+5=11$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

Whakaurua te 2 mō x ki y=2x+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2\times 2+1$$

Whakareatia 2 ki te 2.

$$y=4+1$$

Tāpiri 1 ki te 4.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-2x=1$$

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

$$y-2x=1,5y-7x=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\5&-7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\5&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\5&-7\end{matrix}\right))\left(\begin{matrix}1&-2\\5&-7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\5&-7\end{matrix}\right))\left(\begin{matrix}1\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\5&-7\end{matrix}\right))\left(\begin{matrix}1\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\5&-7\end{matrix}\right))\left(\begin{matrix}1\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{-7-\left(-2\times 5\right)}&-\frac{-2}{-7-\left(-2\times 5\right)}\\-\frac{5}{-7-\left(-2\times 5\right)}&\frac{1}{-7-\left(-2\times 5\right)}\end{matrix}\right)\left(\begin{matrix}1\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{3}&\frac{2}{3}\\-\frac{5}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}1\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{3}+\frac{2}{3}\times 11\\-\frac{5}{3}+\frac{1}{3}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=5,x=2$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-2x=1$$

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.

$$y-2x=1,5y-7x=11$$

Kia ōrite ai a y me 5y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$5y+5\left(-2\right)x=5,5y-7x=11$$

Whakarūnātia.

$$5y-10x=5,5y-7x=11$$

Me tango 5y-7x=11 mai i 5y-10x=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5y-5y-10x+7x=5-11$$

Tāpiri 5y ki te -5y. Ka whakakore atu ngā kupu 5y me -5y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-10x+7x=5-11$$

Tāpiri -10x ki te 7x.

$$-3x=5-11$$

Tāpiri 5 ki te -11.

$$-3x=-6$$

Whakawehea ngā taha e rua ki te -3.

$$x=2$$

Whakaurua te 2 mō x ki 5y-7x=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$5y-7\times 2=11$$

Whakareatia -7 ki te 2.

$$5y-14=11$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$5y=25$$

Whakawehea ngā taha e rua ki te 5.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=2$$