Whakaoti i te {c}{y=-3x+1}{y=x-7}

$$\left. \begin{array} { c } { y = - 3 x + 1 } \\ { y = x - 7 } \end{array} \right.$$

Whakaoti mō y, x

$$x=2$$

$$y=-5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y+3x=1$$

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

$$y-x=-7$$

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.

$$y+3x=1,y-x=-7$$

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

$$y=-3x+1$$

Whakakapia te -3x+1 mō te y ki tērā atu whārite, y-x=-7.

$$-3x+1-x=-7$$

Tāpiri -3x ki te -x.

$$-4x+1=-7$$

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

$$-4x=-8$$

Whakawehea ngā taha e rua ki te -4.

$$x=2$$

Whakaurua te 2 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.

$$y=-3\times 2+1$$

Whakareatia -3 ki te 2.

$$y=-6+1$$

Tāpiri 1 ki te -6.

$$y=-5$$

Kua oti te pūnaha te whakatau.

$$y=-5,x=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y+3x=1$$

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

$$y-x=-7$$

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

$$y+3x=1,y-x=-7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\1&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1\\-7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&3\\1&-1\end{matrix}\right))\left(\begin{matrix}1&3\\1&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\1&-1\end{matrix}\right))\left(\begin{matrix}1\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\1&-1\end{matrix}\right))\left(\begin{matrix}1\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\1&-1\end{matrix}\right))\left(\begin{matrix}1\\-7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-3}&-\frac{3}{-1-3}\\-\frac{1}{-1-3}&\frac{1}{-1-3}\end{matrix}\right)\left(\begin{matrix}1\\-7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{3}{4}\\\frac{1}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}1\\-7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}+\frac{3}{4}\left(-7\right)\\\frac{1}{4}-\frac{1}{4}\left(-7\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-5\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-5,x=2$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y+3x=1$$

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

$$y-x=-7$$

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.

$$y+3x=1,y-x=-7$$

Me tango y-x=-7 mai i y+3x=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+3x+x=1+7$$

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.

$$3x+x=1+7$$

Tāpiri 3x ki te x.

$$4x=1+7$$

Tāpiri 1 ki te 7.

$$4x=8$$

Whakawehea ngā taha e rua ki te 4.

$$x=2$$

Whakaurua te 2 mō x ki y-x=-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-2=-7$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$y=-5$$

Kua oti te pūnaha te whakatau.

$$y=-5,x=2$$