Whakaoti i te {l}{2x+3y=7}{5x+2y=1}

$$\left. \begin{array} { l } { 2 x + 3 y = 7 } \\ { 5 x + 2 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=3$$

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.

$$2x+3y=7,5x+2y=1$$

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

$$2x+3y=7$$

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

$$2x=-3y+7$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+7\right)$$

Whakareatia \frac{1}{2} ki te -3y+7.

$$x=-\frac{3}{2}y+\frac{7}{2}$$

Whakakapia te \frac{-3y+7}{2} mō te x ki tērā atu whārite, 5x+2y=1.

$$5\left(-\frac{3}{2}y+\frac{7}{2}\right)+2y=1$$

Whakareatia 5 ki te \frac{-3y+7}{2}.

$$-\frac{15}{2}y+\frac{35}{2}+2y=1$$

Tāpiri -\frac{15y}{2} ki te 2y.

$$-\frac{11}{2}y+\frac{35}{2}=1$$

Me tango \frac{35}{2} mai i ngā taha e rua o te whārite.

$$-\frac{11}{2}y=-\frac{33}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{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{7}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 3+\frac{7}{2}$$

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

$$x=\frac{-9+7}{2}$$

Tāpiri \frac{7}{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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=3$$

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.

$$2x+3y=7,5x+2y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\5&2\end{matrix}\right))\left(\begin{matrix}2&3\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\5&2\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\5&2\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\5&2\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-3\times 5}&-\frac{3}{2\times 2-3\times 5}\\-\frac{5}{2\times 2-3\times 5}&\frac{2}{2\times 2-3\times 5}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{11}&\frac{3}{11}\\\frac{5}{11}&-\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{11}\times 7+\frac{3}{11}\\\frac{5}{11}\times 7-\frac{2}{11}\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=3$$

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.

$$2x+3y=7,5x+2y=1$$

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.

$$5\times 2x+5\times 3y=5\times 7,2\times 5x+2\times 2y=2$$

Whakarūnātia.

$$10x+15y=35,10x+4y=2$$

Me tango 10x+4y=2 mai i 10x+15y=35 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+15y-4y=35-2$$

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.

$$15y-4y=35-2$$

Tāpiri 15y ki te -4y.

$$11y=35-2$$

Tāpiri 35 ki te -2.

$$11y=33$$

Whakawehea ngā taha e rua ki te 11.

$$y=3$$

Whakaurua te 3 mō y ki 5x+2y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+2\times 3=1$$

Whakareatia 2 ki te 3.

$$5x+6=1$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$5x=-5$$

Whakawehea ngā taha e rua ki te 5.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=3$$