Whakaoti i te {l}{2x+5y=13}{x+7y=-17}

$$\left\{ \begin{array} { l } { 2 x + 5 y = 13 } \\ { x + 7 y = - 17 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{176}{9} = 19\frac{5}{9} \approx 19.555555556$$

$$y = -\frac{47}{9} = -5\frac{2}{9} \approx -5.222222222$$

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.

$$2x+5y=13,x+7y=-17$$

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

$$2x+5y=13$$

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

$$2x=-5y+13$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y+13\right)$$

Whakareatia \frac{1}{2} ki te -5y+13.

$$x=-\frac{5}{2}y+\frac{13}{2}$$

Whakakapia te \frac{-5y+13}{2} mō te x ki tērā atu whārite, x+7y=-17.

$$-\frac{5}{2}y+\frac{13}{2}+7y=-17$$

Tāpiri -\frac{5y}{2} ki te 7y.

$$\frac{9}{2}y+\frac{13}{2}=-17$$

Me tango \frac{13}{2} mai i ngā taha e rua o te whārite.

$$\frac{9}{2}y=-\frac{47}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{47}{9}$$

Whakaurua te -\frac{47}{9} mō y ki x=-\frac{5}{2}y+\frac{13}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}\left(-\frac{47}{9}\right)+\frac{13}{2}$$

Whakareatia -\frac{5}{2} ki te -\frac{47}{9} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{235}{18}+\frac{13}{2}$$

Tāpiri \frac{13}{2} ki te \frac{235}{18} 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.

$$x=\frac{176}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{176}{9},y=-\frac{47}{9}$$

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.

$$2x+5y=13,x+7y=-17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&5\\1&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\-17\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&5\\1&7\end{matrix}\right))\left(\begin{matrix}2&5\\1&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\1&7\end{matrix}\right))\left(\begin{matrix}13\\-17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\1&7\end{matrix}\right))\left(\begin{matrix}13\\-17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\1&7\end{matrix}\right))\left(\begin{matrix}13\\-17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2\times 7-5}&-\frac{5}{2\times 7-5}\\-\frac{1}{2\times 7-5}&\frac{2}{2\times 7-5}\end{matrix}\right)\left(\begin{matrix}13\\-17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{9}&-\frac{5}{9}\\-\frac{1}{9}&\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}13\\-17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{9}\times 13-\frac{5}{9}\left(-17\right)\\-\frac{1}{9}\times 13+\frac{2}{9}\left(-17\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{176}{9}\\-\frac{47}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{176}{9},y=-\frac{47}{9}$$

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.

$$2x+5y=13,x+7y=-17$$

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.

$$2x+5y=13,2x+2\times 7y=2\left(-17\right)$$

Whakarūnātia.

$$2x+5y=13,2x+14y=-34$$

Me tango 2x+14y=-34 mai i 2x+5y=13 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+5y-14y=13+34$$

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.

$$5y-14y=13+34$$

Tāpiri 5y ki te -14y.

$$-9y=13+34$$

Tāpiri 13 ki te 34.

$$-9y=47$$

Whakawehea ngā taha e rua ki te -9.

$$y=-\frac{47}{9}$$

Whakaurua te -\frac{47}{9} mō y ki x+7y=-17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+7\left(-\frac{47}{9}\right)=-17$$

Whakareatia 7 ki te -\frac{47}{9}.

$$x-\frac{329}{9}=-17$$

Me tāpiri \frac{329}{9} ki ngā taha e rua o te whārite.

$$x=\frac{176}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{176}{9},y=-\frac{47}{9}$$