Whakaoti i te {l}{3x-y=19}{2x+7y=5}

$$\left\{ \begin{array} { l } { 3 x - y = 19 } \\ { 2 x + 7 y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=-1$$

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.

$$3x-y=19,2x+7y=5$$

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

$$3x-y=19$$

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

$$3x=y+19$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(y+19\right)$$

Whakareatia \frac{1}{3} ki te y+19.

$$x=\frac{1}{3}y+\frac{19}{3}$$

Whakakapia te \frac{19+y}{3} mō te x ki tērā atu whārite, 2x+7y=5.

$$2\left(\frac{1}{3}y+\frac{19}{3}\right)+7y=5$$

Whakareatia 2 ki te \frac{19+y}{3}.

$$\frac{2}{3}y+\frac{38}{3}+7y=5$$

Tāpiri \frac{2y}{3} ki te 7y.

$$\frac{23}{3}y+\frac{38}{3}=5$$

Me tango \frac{38}{3} mai i ngā taha e rua o te whārite.

$$\frac{23}{3}y=-\frac{23}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{23}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{1}{3}y+\frac{19}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\left(-1\right)+\frac{19}{3}$$

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

$$x=\frac{-1+19}{3}$$

Tāpiri \frac{19}{3} ki te -\frac{1}{3} 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.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-1$$

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.

$$3x-y=19,2x+7y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\2&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}19\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-1\\2&7\end{matrix}\right))\left(\begin{matrix}3&-1\\2&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&7\end{matrix}\right))\left(\begin{matrix}19\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&7\end{matrix}\right))\left(\begin{matrix}19\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&7\end{matrix}\right))\left(\begin{matrix}19\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{3\times 7-\left(-2\right)}&-\frac{-1}{3\times 7-\left(-2\right)}\\-\frac{2}{3\times 7-\left(-2\right)}&\frac{3}{3\times 7-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}19\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{23}&\frac{1}{23}\\-\frac{2}{23}&\frac{3}{23}\end{matrix}\right)\left(\begin{matrix}19\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{23}\times 19+\frac{1}{23}\times 5\\-\frac{2}{23}\times 19+\frac{3}{23}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=-1$$

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.

$$3x-y=19,2x+7y=5$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2\left(-1\right)y=2\times 19,3\times 2x+3\times 7y=3\times 5$$

Whakarūnātia.

$$6x-2y=38,6x+21y=15$$

Me tango 6x+21y=15 mai i 6x-2y=38 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-2y-21y=38-15$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-21y=38-15$$

Tāpiri -2y ki te -21y.

$$-23y=38-15$$

Tāpiri 38 ki te -15.

$$-23y=23$$

Whakawehea ngā taha e rua ki te -23.

$$y=-1$$

Whakaurua te -1 mō y ki 2x+7y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Whakareatia 7 ki te -1.

$$2x-7=5$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$2x=12$$

Whakawehea ngā taha e rua ki te 2.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-1$$