Whakaoti i te {l}{y=4x-5}{y=2x+3}

$$\left. \begin{array} { l } { y = 4 x - 5 } \\ { y = 2 x + 3 } \end{array} \right.$$

Whakaoti mō y, x

$$x=4$$

$$y=11$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-4x=-5$$

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

$$y-2x=3$$

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.

$$y-4x=-5,y-2x=3$$

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

$$y-4x=-5$$

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

$$y=4x-5$$

Whakakapia te 4x-5 mō te y ki tērā atu whārite, y-2x=3.

$$4x-5-2x=3$$

Tāpiri 4x ki te -2x.

$$2x-5=3$$

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

$$2x=8$$

Whakawehea ngā taha e rua ki te 2.

$$x=4$$

Whakaurua te 4 mō x ki y=4x-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=4\times 4-5$$

Whakareatia 4 ki te 4.

$$y=16-5$$

Tāpiri -5 ki te 16.

$$y=11$$

Kua oti te pūnaha te whakatau.

$$y=11,x=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-4x=-5$$

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

$$y-2x=3$$

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

$$y-4x=-5,y-2x=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-4\\1&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-5\\3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-4\\1&-2\end{matrix}\right))\left(\begin{matrix}1&-4\\1&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&-2\end{matrix}\right))\left(\begin{matrix}-5\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&-2\end{matrix}\right))\left(\begin{matrix}-5\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&-2\end{matrix}\right))\left(\begin{matrix}-5\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-\left(-4\right)}&-\frac{-4}{-2-\left(-4\right)}\\-\frac{1}{-2-\left(-4\right)}&\frac{1}{-2-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}-5\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}11\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=11,x=4$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-4x=-5$$

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

$$y-2x=3$$

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.

$$y-4x=-5,y-2x=3$$

Me tango y-2x=3 mai i y-4x=-5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-4x+2x=-5-3$$

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.

$$-4x+2x=-5-3$$

Tāpiri -4x ki te 2x.

$$-2x=-5-3$$

Tāpiri -5 ki te -3.

$$-2x=-8$$

Whakawehea ngā taha e rua ki te -2.

$$x=4$$

Whakaurua te 4 mō x ki y-2x=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-2\times 4=3$$

Whakareatia -2 ki te 4.

$$y-8=3$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$y=11$$

Kua oti te pūnaha te whakatau.

$$y=11,x=4$$