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

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

Whakaoti mō y, x

$$x=\frac{3}{7}\approx 0.428571429$$

$$y = -\frac{8}{7} = -1\frac{1}{7} \approx -1.142857143$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-2x=-2$$

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$y+5x=1$$

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.

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

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

$$y-2x=-2$$

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

$$y=2x-2$$

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

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

Tāpiri 2x ki te 5x.

$$7x-2=1$$

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

$$7x=3$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{3}{7}$$

Whakaurua te \frac{3}{7} mō x ki y=2x-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2\times \frac{3}{7}-2$$

Whakareatia 2 ki te \frac{3}{7}.

$$y=\frac{6}{7}-2$$

Tāpiri -2 ki te \frac{6}{7}.

$$y=-\frac{8}{7}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{8}{7},x=\frac{3}{7}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-2x=-2$$

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$y+5x=1$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\1&5\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{7}\\\frac{3}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-\frac{8}{7},x=\frac{3}{7}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-2x=-2$$

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$y+5x=1$$

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.

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

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

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

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.

$$-2x-5x=-2-1$$

Tāpiri -2x ki te -5x.

$$-7x=-2-1$$

Tāpiri -2 ki te -1.

$$-7x=-3$$

Whakawehea ngā taha e rua ki te -7.

$$x=\frac{3}{7}$$

Whakaurua te \frac{3}{7} mō x ki y+5x=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+5\times \frac{3}{7}=1$$

Whakareatia 5 ki te \frac{3}{7}.

$$y+\frac{15}{7}=1$$

Me tango \frac{15}{7} mai i ngā taha e rua o te whārite.

$$y=-\frac{8}{7}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{8}{7},x=\frac{3}{7}$$