Whakaoti i te {l}{2x-y=13}{-4x-6y=-18}

$$\left. \begin{array} { l } { 2 x - y = 13 } \\ { - 4 x - 6 y = - 18 } \end{array} \right.$$

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.

$$2x-y=13,-4x-6y=-18$$

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

$$2x-y=13$$

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

$$2x=y+13$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(y+13\right)$$

Whakareatia \frac{1}{2} ki te y+13.

$$x=\frac{1}{2}y+\frac{13}{2}$$

Whakakapia te \frac{13+y}{2} mō te x ki tērā atu whārite, -4x-6y=-18.

$$-4\left(\frac{1}{2}y+\frac{13}{2}\right)-6y=-18$$

Whakareatia -4 ki te \frac{13+y}{2}.

$$-2y-26-6y=-18$$

Tāpiri -2y ki te -6y.

$$-8y-26=-18$$

Me tāpiri 26 ki ngā taha e rua o te whārite.

$$-8y=8$$

Whakawehea ngā taha e rua ki te -8.

$$y=-1$$

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

$$x=\frac{1}{2}\left(-1\right)+\frac{13}{2}$$

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

$$x=\frac{-1+13}{2}$$

Tāpiri \frac{13}{2} ki te -\frac{1}{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.

$$x=6$$

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.

$$2x-y=13,-4x-6y=-18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-1\\-4&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\-18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-1\\-4&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-1\\-4&-6\end{matrix}\right))\left(\begin{matrix}2&-1\\-4&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-4&-6\end{matrix}\right))\left(\begin{matrix}13\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-4&-6\end{matrix}\right))\left(\begin{matrix}13\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-4&-6\end{matrix}\right))\left(\begin{matrix}13\\-18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{2\left(-6\right)-\left(-\left(-4\right)\right)}&-\frac{-1}{2\left(-6\right)-\left(-\left(-4\right)\right)}\\-\frac{-4}{2\left(-6\right)-\left(-\left(-4\right)\right)}&\frac{2}{2\left(-6\right)-\left(-\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}13\\-18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}&-\frac{1}{16}\\-\frac{1}{4}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}13\\-18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}\times 13-\frac{1}{16}\left(-18\right)\\-\frac{1}{4}\times 13-\frac{1}{8}\left(-18\right)\end{matrix}\right)$$

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.

$$2x-y=13,-4x-6y=-18$$

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.

$$-4\times 2x-4\left(-1\right)y=-4\times 13,2\left(-4\right)x+2\left(-6\right)y=2\left(-18\right)$$

Whakarūnātia.

$$-8x+4y=-52,-8x-12y=-36$$

Me tango -8x-12y=-36 mai i -8x+4y=-52 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x+4y+12y=-52+36$$

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.

$$4y+12y=-52+36$$

Tāpiri 4y ki te 12y.

$$16y=-52+36$$

Tāpiri -52 ki te 36.

$$16y=-16$$

Whakawehea ngā taha e rua ki te 16.

$$y=-1$$

Whakaurua te -1 mō y ki -4x-6y=-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x-6\left(-1\right)=-18$$

Whakareatia -6 ki te -1.

$$-4x+6=-18$$

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

$$-4x=-24$$

Whakawehea ngā taha e rua ki te -4.

$$x=6$$

Kua oti te pūnaha te whakatau.

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