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

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

Whakaoti mō y, x

$$x=1$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-4x=1$$

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=0$$

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.

$$y-4x=1,y-5x=0$$

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

$$y-4x=1$$

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

$$y=4x+1$$

Whakakapia te 4x+1 mō te y ki tērā atu whārite, y-5x=0.

$$4x+1-5x=0$$

Tāpiri 4x ki te -5x.

$$-x+1=0$$

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

$$-x=-1$$

Whakawehea ngā taha e rua ki te -1.

$$x=1$$

Whakaurua te 1 mō x ki y=4x+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=4+1$$

Tāpiri 1 ki te 4.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-4x=1$$

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=0$$

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

$$y-4x=1,y-5x=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-4\\1&-5\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Tangohia ngā huānga poukapa y me x.

$$y=5,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-4x=1$$

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=0$$

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.

$$y-4x=1,y-5x=0$$

Me tango y-5x=0 mai i y-4x=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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.

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

Tāpiri -4x ki te 5x.

$$x=1$$

Whakaurua te 1 mō x ki y-5x=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-5=0$$

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

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=1$$