Whakaoti i te {l}{2x-3y=15}{x+y=1}

$$\left\{ \begin{array} { l } { 2 x - 3 y = 15 } \\ { x + y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{18}{5} = 3\frac{3}{5} = 3.6$$

$$y = -\frac{13}{5} = -2\frac{3}{5} = -2.6$$

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.

$$2x-3y=15,x+y=1$$

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

$$2x-3y=15$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$2x=3y+15$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y+15\right)$$

Whakareatia \frac{1}{2} ki te 15+3y.

$$x=\frac{3}{2}y+\frac{15}{2}$$

Whakakapia te \frac{15+3y}{2} mō te x ki tērā atu whārite, x+y=1.

$$\frac{3}{2}y+\frac{15}{2}+y=1$$

Tāpiri \frac{3y}{2} ki te y.

$$\frac{5}{2}y+\frac{15}{2}=1$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

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

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.

$$y=-\frac{13}{5}$$

Whakaurua te -\frac{13}{5} mō y ki x=\frac{3}{2}y+\frac{15}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\left(-\frac{13}{5}\right)+\frac{15}{2}$$

Whakareatia \frac{3}{2} ki te -\frac{13}{5} 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.

$$x=-\frac{39}{10}+\frac{15}{2}$$

Tāpiri \frac{15}{2} ki te -\frac{39}{10} 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.

$$x=\frac{18}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{18}{5},y=-\frac{13}{5}$$

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.

$$2x-3y=15,x+y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-3\\1&1\end{matrix}\right))\left(\begin{matrix}2&-3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\1&1\end{matrix}\right))\left(\begin{matrix}15\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\1&1\end{matrix}\right))\left(\begin{matrix}15\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\1&1\end{matrix}\right))\left(\begin{matrix}15\\1\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-3\right)}&-\frac{-3}{2-\left(-3\right)}\\-\frac{1}{2-\left(-3\right)}&\frac{2}{2-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}15\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{3}{5}\\-\frac{1}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}15\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 15+\frac{3}{5}\\-\frac{1}{5}\times 15+\frac{2}{5}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{5}\\-\frac{13}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{18}{5},y=-\frac{13}{5}$$

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.

$$2x-3y=15,x+y=1$$

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.

$$2x-3y=15,2x+2y=2$$

Me tango 2x+2y=2 mai i 2x-3y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-3y-2y=15-2$$

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.

$$-3y-2y=15-2$$

Tāpiri -3y ki te -2y.

$$-5y=15-2$$

Tāpiri 15 ki te -2.

$$-5y=13$$

Whakawehea ngā taha e rua ki te -5.

$$y=-\frac{13}{5}$$

Whakaurua te -\frac{13}{5} mō y ki x+y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\frac{13}{5}=1$$

Me tāpiri \frac{13}{5} ki ngā taha e rua o te whārite.

$$x=\frac{18}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{18}{5},y=-\frac{13}{5}$$