Whakaoti i te {l}{y=4x-2}{y=-x+18}

$$\left. \begin{array} { l } { y = 4 x - 2 } \\ { y = - x + 18 } \end{array} \right.$$

Whakaoti mō y, x

$$x=4$$

$$y=14$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-4x=-2$$

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

$$y+x=18$$

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.

$$y-4x=-2,y+x=18$$

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

$$y-4x=-2$$

Me tāpiri 4x ki ngā taha e rua o te whārite.

$$y=4x-2$$

Whakakapia te 4x-2 mō te y ki tērā atu whārite, y+x=18.

$$4x-2+x=18$$

Tāpiri 4x ki te x.

$$5x-2=18$$

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

$$5x=20$$

Whakawehea ngā taha e rua ki te 5.

$$x=4$$

Whakaurua te 4 mō x ki y=4x-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=4\times 4-2$$

Whakareatia 4 ki te 4.

$$y=16-2$$

Tāpiri -2 ki te 16.

$$y=14$$

Kua oti te pūnaha te whakatau.

$$y=14,x=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-4x=-2$$

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

$$y+x=18$$

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

$$y-4x=-2,y+x=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-4\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-2\\18\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-4\\1&1\end{matrix}\right))\left(\begin{matrix}1&-4\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&1\end{matrix}\right))\left(\begin{matrix}-2\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&1\end{matrix}\right))\left(\begin{matrix}-2\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&1\end{matrix}\right))\left(\begin{matrix}-2\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-4\right)}&-\frac{-4}{1-\left(-4\right)}\\-\frac{1}{1-\left(-4\right)}&\frac{1}{1-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}-2\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{4}{5}\\-\frac{1}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}-2\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\left(-2\right)+\frac{4}{5}\times 18\\-\frac{1}{5}\left(-2\right)+\frac{1}{5}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}14\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=14,x=4$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-4x=-2$$

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

$$y+x=18$$

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.

$$y-4x=-2,y+x=18$$

Me tango y+x=18 mai i y-4x=-2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-4x-x=-2-18$$

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.

$$-4x-x=-2-18$$

Tāpiri -4x ki te -x.

$$-5x=-2-18$$

Tāpiri -2 ki te -18.

$$-5x=-20$$

Whakawehea ngā taha e rua ki te -5.

$$x=4$$

Whakaurua te 4 mō x ki y+x=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+4=18$$

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

$$y=14$$

Kua oti te pūnaha te whakatau.

$$y=14,x=4$$