Whakaoti i te {l}{y=-4/6x-5}{8x+5y=-45}

$$\left\{ \begin{array} { l } { y = - \frac { 4 } { 6 } x - 5 } \\ { 8 x + 5 y = - 45 } \end{array} \right.$$

Whakaoti mō y, x

$$x = -\frac{30}{7} = -4\frac{2}{7} \approx -4.285714286$$

$$y = -\frac{15}{7} = -2\frac{1}{7} \approx -2.142857143$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakahekea te hautanga \frac{4}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y=-\frac{2}{3}x-5$$

Whakakapia te -\frac{2x}{3}-5 mō te y ki tērā atu whārite, 5y+8x=-45.

$$5\left(-\frac{2}{3}x-5\right)+8x=-45$$

Whakareatia 5 ki te -\frac{2x}{3}-5.

$$-\frac{10}{3}x-25+8x=-45$$

Tāpiri -\frac{10x}{3} ki te 8x.

$$\frac{14}{3}x-25=-45$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$\frac{14}{3}x=-20$$

Whakawehea ngā taha e rua o te whārite ki te \frac{14}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{30}{7}$$

Whakaurua te -\frac{30}{7} mō x ki y=-\frac{2}{3}x-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{2}{3}\left(-\frac{30}{7}\right)-5$$

Whakareatia -\frac{2}{3} ki te -\frac{30}{7} 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.

$$y=\frac{20}{7}-5$$

Tāpiri -5 ki te \frac{20}{7}.

$$y=-\frac{15}{7}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{15}{7},x=-\frac{30}{7}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakahekea te hautanga \frac{4}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y=-\frac{2}{3}x-5$$

Me tāpiri te \frac{2}{3}x ki ngā taha e rua.

$$y+\frac{2}{3}x=-5$$

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

$$y+\frac{2}{3}x=-5,5y+8x=-45$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&\frac{2}{3}\\5&8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-5\\-45\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&\frac{2}{3}\\5&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&\frac{2}{3}\\5&8\end{matrix}\right))\left(\begin{matrix}1&\frac{2}{3}\\5&8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{2}{3}\\5&8\end{matrix}\right))\left(\begin{matrix}-5\\-45\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{2}{3}\\5&8\end{matrix}\right))\left(\begin{matrix}-5\\-45\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{2}{3}\\5&8\end{matrix}\right))\left(\begin{matrix}-5\\-45\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{8}{8-\frac{2}{3}\times 5}&-\frac{\frac{2}{3}}{8-\frac{2}{3}\times 5}\\-\frac{5}{8-\frac{2}{3}\times 5}&\frac{1}{8-\frac{2}{3}\times 5}\end{matrix}\right)\left(\begin{matrix}-5\\-45\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{12}{7}&-\frac{1}{7}\\-\frac{15}{14}&\frac{3}{14}\end{matrix}\right)\left(\begin{matrix}-5\\-45\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{12}{7}\left(-5\right)-\frac{1}{7}\left(-45\right)\\-\frac{15}{14}\left(-5\right)+\frac{3}{14}\left(-45\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{7}\\-\frac{30}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-\frac{15}{7},x=-\frac{30}{7}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakahekea te hautanga \frac{4}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y=-\frac{2}{3}x-5$$

Me tāpiri te \frac{2}{3}x ki ngā taha e rua.

$$y+\frac{2}{3}x=-5$$

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.

$$y+\frac{2}{3}x=-5,5y+8x=-45$$

Kia ōrite ai a y me 5y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$5y+5\times \frac{2}{3}x=5\left(-5\right),5y+8x=-45$$

Whakarūnātia.

$$5y+\frac{10}{3}x=-25,5y+8x=-45$$

Me tango 5y+8x=-45 mai i 5y+\frac{10}{3}x=-25 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5y-5y+\frac{10}{3}x-8x=-25+45$$

Tāpiri 5y ki te -5y. Ka whakakore atu ngā kupu 5y me -5y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{10}{3}x-8x=-25+45$$

Tāpiri \frac{10x}{3} ki te -8x.

$$-\frac{14}{3}x=-25+45$$

Tāpiri -25 ki te 45.

$$-\frac{14}{3}x=20$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{14}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{30}{7}$$

Whakaurua te -\frac{30}{7} mō x ki 5y+8x=-45. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$5y+8\left(-\frac{30}{7}\right)=-45$$

Whakareatia 8 ki te -\frac{30}{7}.

$$5y-\frac{240}{7}=-45$$

Me tāpiri \frac{240}{7} ki ngā taha e rua o te whārite.

$$5y=-\frac{75}{7}$$

Whakawehea ngā taha e rua ki te 5.

$$y=-\frac{15}{7}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{15}{7},x=-\frac{30}{7}$$