Whakaoti i te {l}{y=2x-7}{y=-3x-2}

$$\left. \begin{array} { l } { y = 2 x - 7 } \\ { y = - 3 x - 2 } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=-5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=-7$$

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

$$y+3x=-2$$

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.

$$y-2x=-7,y+3x=-2$$

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

$$y-2x=-7$$

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

$$y=2x-7$$

Whakakapia te 2x-7 mō te y ki tērā atu whārite, y+3x=-2.

$$2x-7+3x=-2$$

Tāpiri 2x ki te 3x.

$$5x-7=-2$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$5x=5$$

Whakawehea ngā taha e rua ki te 5.

$$x=1$$

Whakaurua te 1 mō x ki y=2x-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2-7$$

Tāpiri -7 ki te 2.

$$y=-5$$

Kua oti te pūnaha te whakatau.

$$y=-5,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=-7$$

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

$$y+3x=-2$$

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

$$y-2x=-7,y+3x=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\1&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-7\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\1&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\1&3\end{matrix}\right))\left(\begin{matrix}1&-2\\1&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&3\end{matrix}\right))\left(\begin{matrix}-7\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&3\end{matrix}\right))\left(\begin{matrix}-7\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&3\end{matrix}\right))\left(\begin{matrix}-7\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-\left(-2\right)}&-\frac{-2}{3-\left(-2\right)}\\-\frac{1}{3-\left(-2\right)}&\frac{1}{3-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}-7\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}&\frac{2}{5}\\-\frac{1}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}-7\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\left(-7\right)+\frac{2}{5}\left(-2\right)\\-\frac{1}{5}\left(-7\right)+\frac{1}{5}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-5\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-5,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=-7$$

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

$$y+3x=-2$$

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.

$$y-2x=-7,y+3x=-2$$

Me tango y+3x=-2 mai i y-2x=-7 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-2x-3x=-7+2$$

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.

$$-2x-3x=-7+2$$

Tāpiri -2x ki te -3x.

$$-5x=-7+2$$

Tāpiri -7 ki te 2.

$$-5x=-5$$

Whakawehea ngā taha e rua ki te -5.

$$x=1$$

Whakaurua te 1 mō x ki y+3x=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+3=-2$$

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

$$y=-5$$

Kua oti te pūnaha te whakatau.

$$y=-5,x=1$$