Whakaoti i te {l}{y=-2x+1}{y=4x-5}

$$\left\{ \begin{array} { l } { y = - 2 x + 1 } \\ { y = 4 x - 5 } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y+2x=1$$

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

$$y-4x=-5$$

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.

$$y+2x=1,y-4x=-5$$

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

$$y+2x=1$$

Me tango 2x mai i ngā taha e rua o te whārite.

$$y=-2x+1$$

Whakakapia te -2x+1 mō te y ki tērā atu whārite, y-4x=-5.

$$-2x+1-4x=-5$$

Tāpiri -2x ki te -4x.

$$-6x+1=-5$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-6x=-6$$

Whakawehea ngā taha e rua ki te -6.

$$x=1$$

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

$$y=-2+1$$

Tāpiri 1 ki te -2.

$$y=-1$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y+2x=1$$

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

$$y-4x=-5$$

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

$$y+2x=1,y-4x=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

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

$$y+2x=1$$

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

$$y-4x=-5$$

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.

$$y+2x=1,y-4x=-5$$

Me tango y-4x=-5 mai i y+2x=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+2x+4x=1+5$$

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.

$$2x+4x=1+5$$

Tāpiri 2x ki te 4x.

$$6x=1+5$$

Tāpiri 1 ki te 5.

$$6x=6$$

Whakawehea ngā taha e rua ki te 6.

$$x=1$$

Whakaurua te 1 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.

$$y-4=-5$$

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

$$y=-1$$

Kua oti te pūnaha te whakatau.

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