Whakaoti i te {l}{y=x-2}{y=4x+1}

$$\left\{ \begin{array} { l } { y = x - 2 } \\ { y = 4 x + 1 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-1$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-x=-2$$

Whakaarohia te whārite tuarua. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=1$$

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.

$$y-x=-2,y-4x=1$$

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

$$y-x=-2$$

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

$$y=x-2$$

Whakakapia te x-2 mō te y ki tērā atu whārite, y-4x=1.

$$x-2-4x=1$$

Tāpiri x ki te -4x.

$$-3x-2=1$$

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

$$y=-1-2$$

Tāpiri -2 ki te -1.

$$y=-3$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-x=-2$$

Whakaarohia te whārite tuarua. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=1$$

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

$$y-x=-2,y-4x=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-x=-2$$

Whakaarohia te whārite tuarua. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=1$$

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.

$$y-x=-2,y-4x=1$$

Me tango y-4x=1 mai i y-x=-2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-x+4x=-2-1$$

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.

$$-x+4x=-2-1$$

Tāpiri -x ki te 4x.

$$3x=-2-1$$

Tāpiri -2 ki te -1.

$$3x=-3$$

Whakawehea ngā taha e rua ki te 3.

$$x=-1$$

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

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

Whakareatia -4 ki te -1.

$$y+4=1$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$y=-3$$

Kua oti te pūnaha te whakatau.

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