Whakaoti i te {l}{2x+3y=23}{x-2y=-13}

$$\left\{ \begin{array} { l } { 2 x + 3 y = 23 } \\ { x - 2 y = - 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=7$$

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.

$$2x+3y=23,x-2y=-13$$

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

$$2x+3y=23$$

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

$$2x=-3y+23$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+23\right)$$

Whakareatia \frac{1}{2} ki te -3y+23.

$$x=-\frac{3}{2}y+\frac{23}{2}$$

Whakakapia te \frac{-3y+23}{2} mō te x ki tērā atu whārite, x-2y=-13.

$$-\frac{3}{2}y+\frac{23}{2}-2y=-13$$

Tāpiri -\frac{3y}{2} ki te -2y.

$$-\frac{7}{2}y+\frac{23}{2}=-13$$

Me tango \frac{23}{2} mai i ngā taha e rua o te whārite.

$$-\frac{7}{2}y=-\frac{49}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=-\frac{3}{2}y+\frac{23}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 7+\frac{23}{2}$$

Whakareatia -\frac{3}{2} ki te 7.

$$x=\frac{-21+23}{2}$$

Tāpiri \frac{23}{2} ki te -\frac{21}{2} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=7$$

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.

$$2x+3y=23,x-2y=-13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}23\\-13\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\1&-2\end{matrix}\right))\left(\begin{matrix}2&3\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&-2\end{matrix}\right))\left(\begin{matrix}23\\-13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&-2\end{matrix}\right))\left(\begin{matrix}23\\-13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&-2\end{matrix}\right))\left(\begin{matrix}23\\-13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{2\left(-2\right)-3}&-\frac{3}{2\left(-2\right)-3}\\-\frac{1}{2\left(-2\right)-3}&\frac{2}{2\left(-2\right)-3}\end{matrix}\right)\left(\begin{matrix}23\\-13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&\frac{3}{7}\\\frac{1}{7}&-\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}23\\-13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\times 23+\frac{3}{7}\left(-13\right)\\\frac{1}{7}\times 23-\frac{2}{7}\left(-13\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=7$$

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.

$$2x+3y=23,x-2y=-13$$

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.

$$2x+3y=23,2x+2\left(-2\right)y=2\left(-13\right)$$

Whakarūnātia.

$$2x+3y=23,2x-4y=-26$$

Me tango 2x-4y=-26 mai i 2x+3y=23 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+3y+4y=23+26$$

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.

$$3y+4y=23+26$$

Tāpiri 3y ki te 4y.

$$7y=23+26$$

Tāpiri 23 ki te 26.

$$7y=49$$

Whakawehea ngā taha e rua ki te 7.

$$y=7$$

Whakaurua te 7 mō y ki x-2y=-13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-2\times 7=-13$$

Whakareatia -2 ki te 7.

$$x-14=-13$$

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

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=7$$