Whakaoti i te {l}{-2x-6y=-26}{5x+2y=13}

$$\left. \begin{array} { l } { - 2 x - 6 y = - 26 } \\ { 5 x + 2 y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=4$$

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.

$$-2x-6y=-26,5x+2y=13$$

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

$$-2x-6y=-26$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$-2x=6y-26$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}\left(6y-26\right)$$

Whakareatia -\frac{1}{2} ki te 6y-26.

$$x=-3y+13$$

Whakakapia te -3y+13 mō te x ki tērā atu whārite, 5x+2y=13.

$$5\left(-3y+13\right)+2y=13$$

Whakareatia 5 ki te -3y+13.

$$-15y+65+2y=13$$

Tāpiri -15y ki te 2y.

$$-13y+65=13$$

Me tango 65 mai i ngā taha e rua o te whārite.

$$-13y=-52$$

Whakawehea ngā taha e rua ki te -13.

$$y=4$$

Whakaurua te 4 mō y ki x=-3y+13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3\times 4+13$$

Whakareatia -3 ki te 4.

$$x=-12+13$$

Tāpiri 13 ki te -12.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$

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.

$$-2x-6y=-26,5x+2y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&-6\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-26\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&-6\\5&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&-6\\5&2\end{matrix}\right))\left(\begin{matrix}-2&-6\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-6\\5&2\end{matrix}\right))\left(\begin{matrix}-26\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-6\\5&2\end{matrix}\right))\left(\begin{matrix}-26\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-6\\5&2\end{matrix}\right))\left(\begin{matrix}-26\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-2\times 2-\left(-6\times 5\right)}&-\frac{-6}{-2\times 2-\left(-6\times 5\right)}\\-\frac{5}{-2\times 2-\left(-6\times 5\right)}&-\frac{2}{-2\times 2-\left(-6\times 5\right)}\end{matrix}\right)\left(\begin{matrix}-26\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=4$$

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.

$$-2x-6y=-26,5x+2y=13$$

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.

$$5\left(-2\right)x+5\left(-6\right)y=5\left(-26\right),-2\times 5x-2\times 2y=-2\times 13$$

Whakarūnātia.

$$-10x-30y=-130,-10x-4y=-26$$

Me tango -10x-4y=-26 mai i -10x-30y=-130 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-10x+10x-30y+4y=-130+26$$

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.

$$-30y+4y=-130+26$$

Tāpiri -30y ki te 4y.

$$-26y=-130+26$$

Tāpiri -130 ki te 26.

$$-26y=-104$$

Whakawehea ngā taha e rua ki te -26.

$$y=4$$

Whakaurua te 4 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.

$$5x+2\times 4=13$$

Whakareatia 2 ki te 4.

$$5x+8=13$$

Me tango 8 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.

$$x=1,y=4$$