Whakaoti i te {l}{3x-y=3}{7x+2y=20}

$$\left. \begin{array} { l } { 3 x - y = 3 } \\ { 7 x + 2 y = 20 } \end{array} \right.$$

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.

$$3x-y=3,7x+2y=20$$

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

$$3x-y=3$$

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

$$3x=y+3$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=\frac{1}{3}y+1$$

Whakakapia te \frac{y}{3}+1 mō te x ki tērā atu whārite, 7x+2y=20.

$$7\left(\frac{1}{3}y+1\right)+2y=20$$

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

$$\frac{7}{3}y+7+2y=20$$

Tāpiri \frac{7y}{3} ki te 2y.

$$\frac{13}{3}y+7=20$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$\frac{13}{3}y=13$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{3}, 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{1}{3}y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\times 3+1$$

Whakareatia \frac{1}{3} ki te 3.

$$x=1+1$$

Tāpiri 1 ki te 1.

$$x=2$$

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.

$$3x-y=3,7x+2y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-1\\7&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-1\\7&2\end{matrix}\right))\left(\begin{matrix}3&-1\\7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\7&2\end{matrix}\right))\left(\begin{matrix}3\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\7&2\end{matrix}\right))\left(\begin{matrix}3\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\7&2\end{matrix}\right))\left(\begin{matrix}3\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-\left(-7\right)}&-\frac{-1}{3\times 2-\left(-7\right)}\\-\frac{7}{3\times 2-\left(-7\right)}&\frac{3}{3\times 2-\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}3\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}&\frac{1}{13}\\-\frac{7}{13}&\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}3\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}\times 3+\frac{1}{13}\times 20\\-\frac{7}{13}\times 3+\frac{3}{13}\times 20\end{matrix}\right)$$

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.

$$3x-y=3,7x+2y=20$$

Kia ōrite ai a 3x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$7\times 3x+7\left(-1\right)y=7\times 3,3\times 7x+3\times 2y=3\times 20$$

Whakarūnātia.

$$21x-7y=21,21x+6y=60$$

Me tango 21x+6y=60 mai i 21x-7y=21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$21x-21x-7y-6y=21-60$$

Tāpiri 21x ki te -21x. Ka whakakore atu ngā kupu 21x me -21x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-7y-6y=21-60$$

Tāpiri -7y ki te -6y.

$$-13y=21-60$$

Tāpiri 21 ki te -60.

$$-13y=-39$$

Whakawehea ngā taha e rua ki te -13.

$$y=3$$

Whakaurua te 3 mō y ki 7x+2y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+2\times 3=20$$

Whakareatia 2 ki te 3.

$$7x+6=20$$

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

$$7x=14$$

Whakawehea ngā taha e rua ki te 7.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$