Whakaoti i te {l}{4x-y=11}{-2x+3y=-3}

$$\left\{ \begin{array}{l}{ 4 x - y = 11 }\\{ - 2 x + 3 y = - 3 }\end{array} \right.$$

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.

$$4x-y=11,-2x+3y=-3$$

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

$$4x-y=11$$

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

$$4x=y+11$$

Whakawehea ngā taha e rua ki te 4.

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

Whakareatia \frac{1}{4} ki te y+11.

$$x=\frac{1}{4}y+\frac{11}{4}$$

Whakakapia te \frac{11+y}{4} mō te x ki tērā atu whārite, -2x+3y=-3.

$$-2\left(\frac{1}{4}y+\frac{11}{4}\right)+3y=-3$$

Whakareatia -2 ki te \frac{11+y}{4}.

$$-\frac{1}{2}y-\frac{11}{2}+3y=-3$$

Tāpiri -\frac{y}{2} ki te 3y.

$$\frac{5}{2}y-\frac{11}{2}=-3$$

Me tāpiri \frac{11}{2} ki ngā taha e rua o te whārite.

$$\frac{5}{2}y=\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, 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}{4}y+\frac{11}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1+11}{4}$$

Tāpiri \frac{11}{4} ki te \frac{1}{4} 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.

$$4x-y=11,-2x+3y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-1\\-2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\-3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-1\\-2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}4&-1\\-2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}11\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}11\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}11\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4\times 3-\left(-\left(-2\right)\right)}&-\frac{-1}{4\times 3-\left(-\left(-2\right)\right)}\\-\frac{-2}{4\times 3-\left(-\left(-2\right)\right)}&\frac{4}{4\times 3-\left(-\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}11\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}&\frac{1}{10}\\\frac{1}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}11\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}\times 11+\frac{1}{10}\left(-3\right)\\\frac{1}{5}\times 11+\frac{2}{5}\left(-3\right)\end{matrix}\right)$$

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.

$$4x-y=11,-2x+3y=-3$$

Kia ōrite ai a 4x 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 4.

$$-2\times 4x-2\left(-1\right)y=-2\times 11,4\left(-2\right)x+4\times 3y=4\left(-3\right)$$

Whakarūnātia.

$$-8x+2y=-22,-8x+12y=-12$$

Me tango -8x+12y=-12 mai i -8x+2y=-22 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x+2y-12y=-22+12$$

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.

$$2y-12y=-22+12$$

Tāpiri 2y ki te -12y.

$$-10y=-22+12$$

Tāpiri -22 ki te 12.

$$-10y=-10$$

Whakawehea ngā taha e rua ki te -10.

$$y=1$$

Whakaurua te 1 mō y ki -2x+3y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+3=-3$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$-2x=-6$$

Whakawehea ngā taha e rua ki te -2.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$