Whakaoti i te {l}{y=2x-5}{y=-4x+7}

$$\left\{ \begin{array} { l } { y = 2 x - 5 } \\ { y = - 4 x + 7 } \end{array} \right.$$

Whakaoti mō y, x

$$x=2$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-2x=-5$$

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

$$y+4x=7$$

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.

$$y-2x=-5,y+4x=7$$

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

$$y-2x=-5$$

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

$$y=2x-5$$

Whakakapia te 2x-5 mō te y ki tērā atu whārite, y+4x=7.

$$2x-5+4x=7$$

Tāpiri 2x ki te 4x.

$$6x-5=7$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$6x=12$$

Whakawehea ngā taha e rua ki te 6.

$$x=2$$

Whakaurua te 2 mō x ki y=2x-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2\times 2-5$$

Whakareatia 2 ki te 2.

$$y=4-5$$

Tāpiri -5 ki te 4.

$$y=-1$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-2x=-5$$

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

$$y+4x=7$$

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

$$y-2x=-5,y+4x=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\1&4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-5\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\1&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\1&4\end{matrix}\right))\left(\begin{matrix}1&-2\\1&4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&4\end{matrix}\right))\left(\begin{matrix}-5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&4\end{matrix}\right))\left(\begin{matrix}-5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&4\end{matrix}\right))\left(\begin{matrix}-5\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-\left(-2\right)}&-\frac{-2}{4-\left(-2\right)}\\-\frac{1}{4-\left(-2\right)}&\frac{1}{4-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}-5\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&\frac{1}{3}\\-\frac{1}{6}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}-5\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\left(-5\right)+\frac{1}{3}\times 7\\-\frac{1}{6}\left(-5\right)+\frac{1}{6}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-2x=-5$$

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

$$y+4x=7$$

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.

$$y-2x=-5,y+4x=7$$

Me tango y+4x=7 mai i y-2x=-5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-2x-4x=-5-7$$

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.

$$-2x-4x=-5-7$$

Tāpiri -2x ki te -4x.

$$-6x=-5-7$$

Tāpiri -5 ki te -7.

$$-6x=-12$$

Whakawehea ngā taha e rua ki te -6.

$$x=2$$

Whakaurua te 2 mō x ki y+4x=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+4\times 2=7$$

Whakareatia 4 ki te 2.

$$y+8=7$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$y=-1$$

Kua oti te pūnaha te whakatau.

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