Whakaoti i te {l}{2x-3y=4}{5x-y=2}

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

Whakaoti mō x, y

$$x=\frac{2}{13}\approx 0.153846154$$

$$y = -\frac{16}{13} = -1\frac{3}{13} \approx -1.230769231$$

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.

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

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

$$2x-3y=4$$

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

$$2x=3y+4$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y+4\right)$$

Whakareatia \frac{1}{2} ki te 3y+4.

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

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

$$5\left(\frac{3}{2}y+2\right)-y=2$$

Whakareatia 5 ki te \frac{3y}{2}+2.

$$\frac{15}{2}y+10-y=2$$

Tāpiri \frac{15y}{2} ki te -y.

$$\frac{13}{2}y+10=2$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$\frac{13}{2}y=-8$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{16}{13}$$

Whakaurua te -\frac{16}{13} mō y ki x=\frac{3}{2}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\left(-\frac{16}{13}\right)+2$$

Whakareatia \frac{3}{2} ki te -\frac{16}{13} 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.

$$x=-\frac{24}{13}+2$$

Tāpiri 2 ki te -\frac{24}{13}.

$$x=\frac{2}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2}{13},y=-\frac{16}{13}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\5&-1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}&\frac{3}{13}\\-\frac{5}{13}&\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}4\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}\times 4+\frac{3}{13}\times 2\\-\frac{5}{13}\times 4+\frac{2}{13}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}\\-\frac{16}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{2}{13},y=-\frac{16}{13}$$

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.

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

Kia ōrite ai a 2x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$5\times 2x+5\left(-3\right)y=5\times 4,2\times 5x+2\left(-1\right)y=2\times 2$$

Whakarūnātia.

$$10x-15y=20,10x-2y=4$$

Me tango 10x-2y=4 mai i 10x-15y=20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-15y+2y=20-4$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-15y+2y=20-4$$

Tāpiri -15y ki te 2y.

$$-13y=20-4$$

Tāpiri 20 ki te -4.

$$-13y=16$$

Whakawehea ngā taha e rua ki te -13.

$$y=-\frac{16}{13}$$

Whakaurua te -\frac{16}{13} mō y ki 5x-y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-\left(-\frac{16}{13}\right)=2$$

Me tango \frac{16}{13} mai i ngā taha e rua o te whārite.

$$5x=\frac{10}{13}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{2}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2}{13},y=-\frac{16}{13}$$