Whakaoti i te {l}{y=8x-5}{y=-7x+10}

$$\left. \begin{array} { l } { y = 8 x - 5 } \\ { y = - 7 x + 10 } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 8x mai i ngā taha e rua.

$$y-8x=-5$$

Whakaarohia te whārite tuarua. Me tāpiri te 7x ki ngā taha e rua.

$$y+7x=10$$

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.

$$y-8x=-5,y+7x=10$$

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

$$y-8x=-5$$

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

$$y=8x-5$$

Whakakapia te 8x-5 mō te y ki tērā atu whārite, y+7x=10.

$$8x-5+7x=10$$

Tāpiri 8x ki te 7x.

$$15x-5=10$$

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

$$15x=15$$

Whakawehea ngā taha e rua ki te 15.

$$x=1$$

Whakaurua te 1 mō x ki y=8x-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=8-5$$

Tāpiri -5 ki te 8.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 8x mai i ngā taha e rua.

$$y-8x=-5$$

Whakaarohia te whārite tuarua. Me tāpiri te 7x ki ngā taha e rua.

$$y+7x=10$$

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

$$y-8x=-5,y+7x=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-8\\1&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-5\\10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-8\\1&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-8\\1&7\end{matrix}\right))\left(\begin{matrix}1&-8\\1&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&7\end{matrix}\right))\left(\begin{matrix}-5\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&7\end{matrix}\right))\left(\begin{matrix}-5\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-8\\1&7\end{matrix}\right))\left(\begin{matrix}-5\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{7-\left(-8\right)}&-\frac{-8}{7-\left(-8\right)}\\-\frac{1}{7-\left(-8\right)}&\frac{1}{7-\left(-8\right)}\end{matrix}\right)\left(\begin{matrix}-5\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{15}&\frac{8}{15}\\-\frac{1}{15}&\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}-5\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{15}\left(-5\right)+\frac{8}{15}\times 10\\-\frac{1}{15}\left(-5\right)+\frac{1}{15}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}3\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=3,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 8x mai i ngā taha e rua.

$$y-8x=-5$$

Whakaarohia te whārite tuarua. Me tāpiri te 7x ki ngā taha e rua.

$$y+7x=10$$

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.

$$y-8x=-5,y+7x=10$$

Me tango y+7x=10 mai i y-8x=-5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-8x-7x=-5-10$$

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.

$$-8x-7x=-5-10$$

Tāpiri -8x ki te -7x.

$$-15x=-5-10$$

Tāpiri -5 ki te -10.

$$-15x=-15$$

Whakawehea ngā taha e rua ki te -15.

$$x=1$$

Whakaurua te 1 mō x ki y+7x=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+7=10$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=1$$