Whakaoti i te {l}{y=-4/5x-9}{y=-8x/3-15}

$$\left\{ \begin{array} { l } { y = \frac { - 4 } { 5 } x - 9 } \\ { y = - \frac { 8 x } { 3 } - 15 } \end{array} \right.$$

Whakaoti mō y, x

$$x = -\frac{45}{14} = -3\frac{3}{14} \approx -3.214285714$$

$$y = -\frac{45}{7} = -6\frac{3}{7} \approx -6.428571429$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Ka taea te hautanga \frac{-4}{5} te tuhi anō ko -\frac{4}{5} mā te tango i te tohu tōraro.

$$y=-\frac{4}{5}x-9$$

Whakakapia te -\frac{4x}{5}-9 mō te y ki tērā atu whārite, 3y+8x=-45.

$$3\left(-\frac{4}{5}x-9\right)+8x=-45$$

Whakareatia 3 ki te -\frac{4x}{5}-9.

$$-\frac{12}{5}x-27+8x=-45$$

Tāpiri -\frac{12x}{5} ki te 8x.

$$\frac{28}{5}x-27=-45$$

Me tāpiri 27 ki ngā taha e rua o te whārite.

$$\frac{28}{5}x=-18$$

Whakawehea ngā taha e rua o te whārite ki te \frac{28}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{45}{14}$$

Whakaurua te -\frac{45}{14} mō x ki y=-\frac{4}{5}x-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{4}{5}\left(-\frac{45}{14}\right)-9$$

Whakareatia -\frac{4}{5} ki te -\frac{45}{14} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=\frac{18}{7}-9$$

Tāpiri -9 ki te \frac{18}{7}.

$$y=-\frac{45}{7}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{45}{7},x=-\frac{45}{14}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Ka taea te hautanga \frac{-4}{5} te tuhi anō ko -\frac{4}{5} mā te tango i te tohu tōraro.

$$y=-\frac{4}{5}x-9$$

Me tāpiri te \frac{4}{5}x ki ngā taha e rua.

$$y+\frac{4}{5}x=-9$$

Whakaarohia te whārite tuarua. Me tāpiri te \frac{8x}{3} ki ngā taha e rua.

$$y+\frac{8x}{3}=-15$$

Whakareatia ngā taha e rua o te whārite ki te 3.

$$3y+8x=-45$$

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

$$y+\frac{4}{5}x=-9,3y+8x=-45$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&\frac{4}{5}\\3&8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-9\\-45\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&\frac{4}{5}\\3&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&\frac{4}{5}\\3&8\end{matrix}\right))\left(\begin{matrix}1&\frac{4}{5}\\3&8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{4}{5}\\3&8\end{matrix}\right))\left(\begin{matrix}-9\\-45\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{4}{5}\\3&8\end{matrix}\right))\left(\begin{matrix}-9\\-45\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{4}{5}\\3&8\end{matrix}\right))\left(\begin{matrix}-9\\-45\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{8}{8-\frac{4}{5}\times 3}&-\frac{\frac{4}{5}}{8-\frac{4}{5}\times 3}\\-\frac{3}{8-\frac{4}{5}\times 3}&\frac{1}{8-\frac{4}{5}\times 3}\end{matrix}\right)\left(\begin{matrix}-9\\-45\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{10}{7}&-\frac{1}{7}\\-\frac{15}{28}&\frac{5}{28}\end{matrix}\right)\left(\begin{matrix}-9\\-45\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{10}{7}\left(-9\right)-\frac{1}{7}\left(-45\right)\\-\frac{15}{28}\left(-9\right)+\frac{5}{28}\left(-45\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{45}{7}\\-\frac{45}{14}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-\frac{45}{7},x=-\frac{45}{14}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Ka taea te hautanga \frac{-4}{5} te tuhi anō ko -\frac{4}{5} mā te tango i te tohu tōraro.

$$y=-\frac{4}{5}x-9$$

Me tāpiri te \frac{4}{5}x ki ngā taha e rua.

$$y+\frac{4}{5}x=-9$$

Whakaarohia te whārite tuarua. Me tāpiri te \frac{8x}{3} ki ngā taha e rua.

$$y+\frac{8x}{3}=-15$$

Whakareatia ngā taha e rua o te whārite ki te 3.

$$3y+8x=-45$$

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.

$$y+\frac{4}{5}x=-9,3y+8x=-45$$

Kia ōrite ai a y me 3y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$3y+3\times \frac{4}{5}x=3\left(-9\right),3y+8x=-45$$

Whakarūnātia.

$$3y+\frac{12}{5}x=-27,3y+8x=-45$$

Me tango 3y+8x=-45 mai i 3y+\frac{12}{5}x=-27 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3y-3y+\frac{12}{5}x-8x=-27+45$$

Tāpiri 3y ki te -3y. Ka whakakore atu ngā kupu 3y me -3y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{12}{5}x-8x=-27+45$$

Tāpiri \frac{12x}{5} ki te -8x.

$$-\frac{28}{5}x=-27+45$$

Tāpiri -27 ki te 45.

$$-\frac{28}{5}x=18$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{28}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{45}{14}$$

Whakaurua te -\frac{45}{14} mō x ki 3y+8x=-45. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3y+8\left(-\frac{45}{14}\right)=-45$$

Whakareatia 8 ki te -\frac{45}{14}.

$$3y-\frac{180}{7}=-45$$

Me tāpiri \frac{180}{7} ki ngā taha e rua o te whārite.

$$3y=-\frac{135}{7}$$

Whakawehea ngā taha e rua ki te 3.

$$y=-\frac{45}{7}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{45}{7},x=-\frac{45}{14}$$