Whakaoti i te {l}{y=3x+1}{y=6x+4}

$$\left. \begin{array} { l } { y = 3 x + 1 } \\ { y = 6 x + 4 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-1$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=1$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=4$$

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.

$$y-3x=1,y-6x=4$$

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

$$y-3x=1$$

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

$$y=3x+1$$

Whakakapia te 3x+1 mō te y ki tērā atu whārite, y-6x=4.

$$3x+1-6x=4$$

Tāpiri 3x ki te -6x.

$$-3x+1=4$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-3x=3$$

Whakawehea ngā taha e rua ki te -3.

$$x=-1$$

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

$$y=3\left(-1\right)+1$$

Whakareatia 3 ki te -1.

$$y=-3+1$$

Tāpiri 1 ki te -3.

$$y=-2$$

Kua oti te pūnaha te whakatau.

$$y=-2,x=-1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=1$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=4$$

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

$$y-3x=1,y-6x=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\1&-6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-3\\1&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-3\\1&-6\end{matrix}\right))\left(\begin{matrix}1&-3\\1&-6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&-6\end{matrix}\right))\left(\begin{matrix}1\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&-6\end{matrix}\right))\left(\begin{matrix}1\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&-6\end{matrix}\right))\left(\begin{matrix}1\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-6-\left(-3\right)}&-\frac{-3}{-6-\left(-3\right)}\\-\frac{1}{-6-\left(-3\right)}&\frac{1}{-6-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}1\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=-2,x=-1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=1$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=4$$

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.

$$y-3x=1,y-6x=4$$

Me tango y-6x=4 mai i y-3x=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-3x+6x=1-4$$

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

$$-3x+6x=1-4$$

Tāpiri -3x ki te 6x.

$$3x=1-4$$

Tāpiri 1 ki te -4.

$$3x=-3$$

Whakawehea ngā taha e rua ki te 3.

$$x=-1$$

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

$$y-6\left(-1\right)=4$$

Whakareatia -6 ki te -1.

$$y+6=4$$

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

$$y=-2$$

Kua oti te pūnaha te whakatau.

$$y=-2,x=-1$$