Whakaoti i te {l}{3x-7y=2}{-5x+2y=-13}

$$\left. \begin{array} { l } { 3 x - 7 y = 2 } \\ { - 5 x + 2 y = - 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=1$$

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.

$$3x-7y=2,-5x+2y=-13$$

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

$$3x-7y=2$$

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

$$3x=7y+2$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te \frac{7y+2}{3} mō te x ki tērā atu whārite, -5x+2y=-13.

$$-5\left(\frac{7}{3}y+\frac{2}{3}\right)+2y=-13$$

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

$$-\frac{35}{3}y-\frac{10}{3}+2y=-13$$

Tāpiri -\frac{35y}{3} ki te 2y.

$$-\frac{29}{3}y-\frac{10}{3}=-13$$

Me tāpiri \frac{10}{3} ki ngā taha e rua o te whārite.

$$-\frac{29}{3}y=-\frac{29}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{29}{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{7}{3}y+\frac{2}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Tāpiri \frac{2}{3} ki te \frac{7}{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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$

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.

$$3x-7y=2,-5x+2y=-13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-7\\-5&2\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-7\\-5&2\end{matrix}\right))\left(\begin{matrix}2\\-13\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{29}&-\frac{7}{29}\\-\frac{5}{29}&-\frac{3}{29}\end{matrix}\right)\left(\begin{matrix}2\\-13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{29}\times 2-\frac{7}{29}\left(-13\right)\\-\frac{5}{29}\times 2-\frac{3}{29}\left(-13\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=1$$

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.

$$3x-7y=2,-5x+2y=-13$$

Kia ōrite ai a 3x 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 3.

$$-5\times 3x-5\left(-7\right)y=-5\times 2,3\left(-5\right)x+3\times 2y=3\left(-13\right)$$

Whakarūnātia.

$$-15x+35y=-10,-15x+6y=-39$$

Me tango -15x+6y=-39 mai i -15x+35y=-10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-15x+15x+35y-6y=-10+39$$

Tāpiri -15x ki te 15x. Ka whakakore atu ngā kupu -15x me 15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$35y-6y=-10+39$$

Tāpiri 35y ki te -6y.

$$29y=-10+39$$

Tāpiri -10 ki te 39.

$$29y=29$$

Whakawehea ngā taha e rua ki te 29.

$$y=1$$

Whakaurua te 1 mō y ki -5x+2y=-13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+2=-13$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$-5x=-15$$

Whakawehea ngā taha e rua ki te -5.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$