Whakaoti i te {l}{2x-3y=13}{5x-y=13}

$$\left. \begin{array} { l } { 2 x - 3 y = 13 } \\ { 5 x - y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-3$$

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.

$$2x-3y=13,5x-y=13$$

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

$$2x-3y=13$$

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

$$2x=3y+13$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y+13\right)$$

Whakareatia \frac{1}{2} ki te 3y+13.

$$x=\frac{3}{2}y+\frac{13}{2}$$

Whakakapia te \frac{3y+13}{2} mō te x ki tērā atu whārite, 5x-y=13.

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

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

$$\frac{15}{2}y+\frac{65}{2}-y=13$$

Tāpiri \frac{15y}{2} ki te -y.

$$\frac{13}{2}y+\frac{65}{2}=13$$

Me tango \frac{65}{2} mai i ngā taha e rua o te whārite.

$$\frac{13}{2}y=-\frac{39}{2}$$

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.

$$y=-3$$

Whakaurua te -3 mō y ki x=\frac{3}{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.

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

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

$$x=\frac{-9+13}{2}$$

Tāpiri \frac{13}{2} ki te -\frac{9}{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.

$$x=2$$

Kua oti te pūnaha te whakatau.

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

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.

$$2x-3y=13,5x-y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right))\left(\begin{matrix}13\\13\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$2x-3y=13,5x-y=13$$

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.

$$5\times 2x+5\left(-3\right)y=5\times 13,2\times 5x+2\left(-1\right)y=2\times 13$$

Whakarūnātia.

$$10x-15y=65,10x-2y=26$$

Me tango 10x-2y=26 mai i 10x-15y=65 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-15y+2y=65-26$$

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.

$$-15y+2y=65-26$$

Tāpiri -15y ki te 2y.

$$-13y=65-26$$

Tāpiri 65 ki te -26.

$$-13y=39$$

Whakawehea ngā taha e rua ki te -13.

$$y=-3$$

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

$$5x-\left(-3\right)=13$$

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

$$5x=10$$

Whakawehea ngā taha e rua ki te 5.

$$x=2$$

Kua oti te pūnaha te whakatau.

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