Whakaoti i te {c}{2x+3y=-10}{x+4y=5}

$$\left. \begin{array} { c } { 2 x + 3 y = - 10 } \\ { x + 4 y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-11$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$2x+3y=-10,x+4y=5$$

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

$$2x+3y=-10$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y-10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y-10\right)$$

Whakareatia \frac{1}{2} ki te -3y-10.

$$x=-\frac{3}{2}y-5$$

Whakakapia te -\frac{3y}{2}-5 mō te x ki tērā atu whārite, x+4y=5.

$$-\frac{3}{2}y-5+4y=5$$

Tāpiri -\frac{3y}{2} ki te 4y.

$$\frac{5}{2}y-5=5$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$\frac{5}{2}y=10$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

Whakaurua te 4 mō y ki x=-\frac{3}{2}y-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 4-5$$

Whakareatia -\frac{3}{2} ki te 4.

$$x=-6-5$$

Tāpiri -5 ki te -6.

$$x=-11$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$2x+3y=-10,x+4y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\1&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\1&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\1&4\end{matrix}\right))\left(\begin{matrix}2&3\\1&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&4\end{matrix}\right))\left(\begin{matrix}-10\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&4\end{matrix}\right))\left(\begin{matrix}-10\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&4\end{matrix}\right))\left(\begin{matrix}-10\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{2\times 4-3}&-\frac{3}{2\times 4-3}\\-\frac{1}{2\times 4-3}&\frac{2}{2\times 4-3}\end{matrix}\right)\left(\begin{matrix}-10\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5}&-\frac{3}{5}\\-\frac{1}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}-10\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5}\left(-10\right)-\frac{3}{5}\times 5\\-\frac{1}{5}\left(-10\right)+\frac{2}{5}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

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.

$$2x+3y=-10,x+4y=5$$

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

$$2x+3y=-10,2x+2\times 4y=2\times 5$$

Whakarūnātia.

$$2x+3y=-10,2x+8y=10$$

Me tango 2x+8y=10 mai i 2x+3y=-10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+3y-8y=-10-10$$

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

$$3y-8y=-10-10$$

Tāpiri 3y ki te -8y.

$$-5y=-10-10$$

Tāpiri -10 ki te -10.

$$-5y=-20$$

Whakawehea ngā taha e rua ki te -5.

$$y=4$$

Whakaurua te 4 mō y ki x+4y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+4\times 4=5$$

Whakareatia 4 ki te 4.

$$x+16=5$$

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

$$x=-11$$

Kua oti te pūnaha te whakatau.

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