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

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

Whakaoti mō x, y

$$x=1$$

$$y=1$$

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.

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

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

$$3x+y=4$$

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

$$3x=-y+4$$

Whakawehea ngā taha e rua ki te 3.

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

Whakareatia \frac{1}{3} ki te -y+4.

$$x=-\frac{1}{3}y+\frac{4}{3}$$

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

$$-\frac{1}{3}y+\frac{4}{3}-3y=-2$$

Tāpiri -\frac{y}{3} ki te -3y.

$$-\frac{10}{3}y+\frac{4}{3}=-2$$

Me tango \frac{4}{3} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te -\frac{10}{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{1}{3}y+\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{-1+4}{3}$$

Tāpiri \frac{4}{3} ki te -\frac{1}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}&\frac{1}{10}\\\frac{1}{10}&-\frac{3}{10}\end{matrix}\right)\left(\begin{matrix}4\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}\times 4+\frac{1}{10}\left(-2\right)\\\frac{1}{10}\times 4-\frac{3}{10}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=1$$

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.

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

Kia ōrite ai a 3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

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

Whakarūnātia.

$$3x+y=4,3x-9y=-6$$

Me tango 3x-9y=-6 mai i 3x+y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+y+9y=4+6$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y+9y=4+6$$

Tāpiri y ki te 9y.

$$10y=4+6$$

Tāpiri 4 ki te 6.

$$10y=10$$

Whakawehea ngā taha e rua ki te 10.

$$y=1$$

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

$$x-3=-2$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$