Whakaoti i te {l}{3x+y=11}{-4x-y=11}

$$\left\{ \begin{array} { l } { 3 x + y = 11 } \\ { - 4 x - y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-22$$

$$y=77$$

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.

$$3x+y=11,-4x-y=11$$

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

$$3x+y=11$$

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

$$3x=-y+11$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-y+11\right)$$

Whakareatia \frac{1}{3} ki te -y+11.

$$x=-\frac{1}{3}y+\frac{11}{3}$$

Whakakapia te \frac{-y+11}{3} mō te x ki tērā atu whārite, -4x-y=11.

$$-4\left(-\frac{1}{3}y+\frac{11}{3}\right)-y=11$$

Whakareatia -4 ki te \frac{-y+11}{3}.

$$\frac{4}{3}y-\frac{44}{3}-y=11$$

Tāpiri \frac{4y}{3} ki te -y.

$$\frac{1}{3}y-\frac{44}{3}=11$$

Me tāpiri \frac{44}{3} ki ngā taha e rua o te whārite.

$$\frac{1}{3}y=\frac{77}{3}$$

Me whakarea ngā taha e rua ki te 3.

$$y=77$$

Whakaurua te 77 mō y ki x=-\frac{1}{3}y+\frac{11}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times 77+\frac{11}{3}$$

Whakareatia -\frac{1}{3} ki te 77.

$$x=\frac{-77+11}{3}$$

Tāpiri \frac{11}{3} ki te -\frac{77}{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.

$$x=-22$$

Kua oti te pūnaha te whakatau.

$$x=-22,y=77$$

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.

$$3x+y=11,-4x-y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\-4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\-4&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&1\\-4&-1\end{matrix}\right))\left(\begin{matrix}3&1\\-4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\-4&-1\end{matrix}\right))\left(\begin{matrix}11\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\-4&-1\end{matrix}\right))\left(\begin{matrix}11\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\-4&-1\end{matrix}\right))\left(\begin{matrix}11\\11\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-\left(-4\right)}&-\frac{1}{3\left(-1\right)-\left(-4\right)}\\-\frac{-4}{3\left(-1\right)-\left(-4\right)}&\frac{3}{3\left(-1\right)-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}11\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&-1\\4&3\end{matrix}\right)\left(\begin{matrix}11\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-11-11\\4\times 11+3\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-22\\77\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-22,y=77$$

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.

$$3x+y=11,-4x-y=11$$

Kia ōrite ai a 3x 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 3.

$$-4\times 3x-4y=-4\times 11,3\left(-4\right)x+3\left(-1\right)y=3\times 11$$

Whakarūnātia.

$$-12x-4y=-44,-12x-3y=33$$

Me tango -12x-3y=33 mai i -12x-4y=-44 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x-4y+3y=-44-33$$

Tāpiri -12x ki te 12x. Ka whakakore atu ngā kupu -12x me 12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y+3y=-44-33$$

Tāpiri -4y ki te 3y.

$$-y=-44-33$$

Tāpiri -44 ki te -33.

$$-y=-77$$

Whakawehea ngā taha e rua ki te -1.

$$y=77$$

Whakaurua te 77 mō y ki -4x-y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x-77=11$$

Me tāpiri 77 ki ngā taha e rua o te whārite.

$$-4x=88$$

Whakawehea ngā taha e rua ki te -4.

$$x=-22$$

Kua oti te pūnaha te whakatau.

$$x=-22,y=77$$