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

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

Whakaoti mō y, x

$$x=0$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te \frac{x}{3} mai i ngā taha e rua.

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

Whakareatia ngā taha e rua o te whārite ki te 3.

$$3y-x=-9$$

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.

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

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

$$3y-x=-9$$

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

$$3y=x-9$$

Whakawehea ngā taha e rua ki te 3.

$$y=\frac{1}{3}\left(x-9\right)$$

Whakareatia \frac{1}{3} ki te x-9.

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

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

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

Tāpiri \frac{x}{3} ki te 4x.

$$\frac{13}{3}x-3=-3$$

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

$$\frac{13}{3}x=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=0$$

Whakaurua te 0 mō x ki y=\frac{1}{3}x-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-3$$

Kua oti te pūnaha te whakatau.

$$y=-3,x=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te \frac{x}{3} mai i ngā taha e rua.

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

Whakareatia ngā taha e rua o te whārite ki te 3.

$$3y-x=-9$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\1&4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-9\\-3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-1\\1&4\end{matrix}\right))\left(\begin{matrix}3&-1\\1&4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&4\end{matrix}\right))\left(\begin{matrix}-9\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&4\end{matrix}\right))\left(\begin{matrix}-9\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&4\end{matrix}\right))\left(\begin{matrix}-9\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3\times 4-\left(-1\right)}&-\frac{-1}{3\times 4-\left(-1\right)}\\-\frac{1}{3\times 4-\left(-1\right)}&\frac{3}{3\times 4-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}-9\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{13}&\frac{1}{13}\\-\frac{1}{13}&\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}-9\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{13}\left(-9\right)+\frac{1}{13}\left(-3\right)\\-\frac{1}{13}\left(-9\right)+\frac{3}{13}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-3\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-3,x=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te \frac{x}{3} mai i ngā taha e rua.

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

Whakareatia ngā taha e rua o te whārite ki te 3.

$$3y-x=-9$$

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.

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

Kia ōrite ai a 3y me y, 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.

$$3y-x=-9,3y+3\times 4x=3\left(-3\right)$$

Whakarūnātia.

$$3y-x=-9,3y+12x=-9$$

Me tango 3y+12x=-9 mai i 3y-x=-9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3y-3y-x-12x=-9+9$$

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

$$-x-12x=-9+9$$

Tāpiri -x ki te -12x.

$$-13x=-9+9$$

Tāpiri -9 ki te 9.

$$-13x=0$$

Whakawehea ngā taha e rua ki te -13.

$$x=0$$

Whakaurua te 0 mō x ki y+4x=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-3$$

Kua oti te pūnaha te whakatau.

$$y=-3,x=0$$