Whakaoti i te {l}{2x+3y=8}{x-y=10}

$$\left. \begin{array} { l } { 2 x + 3 y = 8 } \\ { x - y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{38}{5} = 7\frac{3}{5} = 7.6$$

$$y = -\frac{12}{5} = -2\frac{2}{5} = -2.4$$

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.

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

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

$$2x+3y=8$$

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

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

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+8\right)$$

Whakareatia \frac{1}{2} ki te -3y+8.

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

Whakakapia te -\frac{3y}{2}+4 mō te x ki tērā atu whārite, x-y=10.

$$-\frac{3}{2}y+4-y=10$$

Tāpiri -\frac{3y}{2} ki te -y.

$$-\frac{5}{2}y+4=10$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-\frac{5}{2}y=6$$

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.

$$y=-\frac{12}{5}$$

Whakaurua te -\frac{12}{5} mō y ki x=-\frac{3}{2}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\left(-\frac{12}{5}\right)+4$$

Whakareatia -\frac{3}{2} ki te -\frac{12}{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.

$$x=\frac{18}{5}+4$$

Tāpiri 4 ki te \frac{18}{5}.

$$x=\frac{38}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{38}{5},y=-\frac{12}{5}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\1&-1\end{matrix}\right))\left(\begin{matrix}2&3\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&-1\end{matrix}\right))\left(\begin{matrix}8\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&-1\end{matrix}\right))\left(\begin{matrix}8\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&-1\end{matrix}\right))\left(\begin{matrix}8\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-3}&-\frac{3}{2\left(-1\right)-3}\\-\frac{1}{2\left(-1\right)-3}&\frac{2}{2\left(-1\right)-3}\end{matrix}\right)\left(\begin{matrix}8\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{3}{5}\\\frac{1}{5}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}8\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 8+\frac{3}{5}\times 10\\\frac{1}{5}\times 8-\frac{2}{5}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{38}{5}\\-\frac{12}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{38}{5},y=-\frac{12}{5}$$

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.

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

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.

$$2x+3y=8,2x+2\left(-1\right)y=2\times 10$$

Whakarūnātia.

$$2x+3y=8,2x-2y=20$$

Me tango 2x-2y=20 mai i 2x+3y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+3y+2y=8-20$$

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.

$$3y+2y=8-20$$

Tāpiri 3y ki te 2y.

$$5y=8-20$$

Tāpiri 8 ki te -20.

$$5y=-12$$

Whakawehea ngā taha e rua ki te 5.

$$y=-\frac{12}{5}$$

Whakaurua te -\frac{12}{5} mō y ki x-y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\left(-\frac{12}{5}\right)=10$$

Me tango \frac{12}{5} mai i ngā taha e rua o te whārite.

$$x=\frac{38}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{38}{5},y=-\frac{12}{5}$$