Whakaoti i te {l}{2x-5y=7}{4x+3y=1}

$$\left\{ \begin{array} { l } { 2 x - 5 y = 7 } \\ { 4 x + 3 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

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.

$$2x-5y=7,4x+3y=1$$

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

$$2x-5y=7$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$2x=5y+7$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te 5y+7.

$$x=\frac{5}{2}y+\frac{7}{2}$$

Whakakapia te \frac{5y+7}{2} mō te x ki tērā atu whārite, 4x+3y=1.

$$4\left(\frac{5}{2}y+\frac{7}{2}\right)+3y=1$$

Whakareatia 4 ki te \frac{5y+7}{2}.

$$10y+14+3y=1$$

Tāpiri 10y ki te 3y.

$$13y+14=1$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$13y=-13$$

Whakawehea ngā taha e rua ki te 13.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{5}{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.

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

Whakareatia \frac{5}{2} ki te -1.

$$x=\frac{-5+7}{2}$$

Tāpiri \frac{7}{2} ki te -\frac{5}{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.

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

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.

$$2x-5y=7,4x+3y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-5\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-5\\4&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-5\\4&3\end{matrix}\right))\left(\begin{matrix}2&-5\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\4&3\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\4&3\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\4&3\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te 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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-\left(-5\times 4\right)}&-\frac{-5}{2\times 3-\left(-5\times 4\right)}\\-\frac{4}{2\times 3-\left(-5\times 4\right)}&\frac{2}{2\times 3-\left(-5\times 4\right)}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{26}&\frac{5}{26}\\-\frac{2}{13}&\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{26}\times 7+\frac{5}{26}\\-\frac{2}{13}\times 7+\frac{1}{13}\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$2x-5y=7,4x+3y=1$$

Kia ōrite ai a 2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$4\times 2x+4\left(-5\right)y=4\times 7,2\times 4x+2\times 3y=2$$

Whakarūnātia.

$$8x-20y=28,8x+6y=2$$

Me tango 8x+6y=2 mai i 8x-20y=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-20y-6y=28-2$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-20y-6y=28-2$$

Tāpiri -20y ki te -6y.

$$-26y=28-2$$

Tāpiri 28 ki te -2.

$$-26y=26$$

Whakawehea ngā taha e rua ki te -26.

$$y=-1$$

Whakaurua te -1 mō y ki 4x+3y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+3\left(-1\right)=1$$

Whakareatia 3 ki te -1.

$$4x-3=1$$

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

$$4x=4$$

Whakawehea ngā taha e rua ki te 4.

$$x=1$$

Kua oti te pūnaha te whakatau.

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