Whakaoti i te {l}{y=-x-7}{5y+3x=-13}

$$\left. \begin{array} { l } { y = - x - 7 } \\ { 5 y + 3 x = - 13 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-11$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te x ki ngā taha e rua.

$$y+x=-7$$

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.

$$y+x=-7,5y+3x=-13$$

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

$$y+x=-7$$

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

$$y=-x-7$$

Whakakapia te -x-7 mō te y ki tērā atu whārite, 5y+3x=-13.

$$5\left(-x-7\right)+3x=-13$$

Whakareatia 5 ki te -x-7.

$$-5x-35+3x=-13$$

Tāpiri -5x ki te 3x.

$$-2x-35=-13$$

Me tāpiri 35 ki ngā taha e rua o te whārite.

$$-2x=22$$

Whakawehea ngā taha e rua ki te -2.

$$x=-11$$

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

$$y=-\left(-11\right)-7$$

Whakareatia -1 ki te -11.

$$y=11-7$$

Tāpiri -7 ki te 11.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$y=4,x=-11$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te x ki ngā taha e rua.

$$y+x=-7$$

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

$$y+x=-7,5y+3x=-13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\5&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-7\\-13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\5&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\5&3\end{matrix}\right))\left(\begin{matrix}1&1\\5&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\5&3\end{matrix}\right))\left(\begin{matrix}-7\\-13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\5&3\end{matrix}\right))\left(\begin{matrix}-7\\-13\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\5&3\end{matrix}\right))\left(\begin{matrix}-7\\-13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-5}&-\frac{1}{3-5}\\-\frac{5}{3-5}&\frac{1}{3-5}\end{matrix}\right)\left(\begin{matrix}-7\\-13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2}&\frac{1}{2}\\\frac{5}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}-7\\-13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2}\left(-7\right)+\frac{1}{2}\left(-13\right)\\\frac{5}{2}\left(-7\right)-\frac{1}{2}\left(-13\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}4\\-11\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=4,x=-11$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te x ki ngā taha e rua.

$$y+x=-7$$

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.

$$y+x=-7,5y+3x=-13$$

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.

$$5y+5x=5\left(-7\right),5y+3x=-13$$

Whakarūnātia.

$$5y+5x=-35,5y+3x=-13$$

Me tango 5y+3x=-13 mai i 5y+5x=-35 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5y-5y+5x-3x=-35+13$$

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.

$$5x-3x=-35+13$$

Tāpiri 5x ki te -3x.

$$2x=-35+13$$

Tāpiri -35 ki te 13.

$$2x=-22$$

Whakawehea ngā taha e rua ki te 2.

$$x=-11$$

Whakaurua te -11 mō x ki 5y+3x=-13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$5y+3\left(-11\right)=-13$$

Whakareatia 3 ki te -11.

$$5y-33=-13$$

Me tāpiri 33 ki ngā taha e rua o te whārite.

$$5y=20$$

Whakawehea ngā taha e rua ki te 5.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$y=4,x=-11$$