Whakaoti i te {l}{5x+2y=-6}{2x+5y=8}

$$\left\{ \begin{array} { l } { 5 x + 2 y = - 6 } \\ { 2 x + 5 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{46}{21} = -2\frac{4}{21} \approx -2.19047619$$

$$y = \frac{52}{21} = 2\frac{10}{21} \approx 2.476190476$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$5x+2y=-6,2x+5y=8$$

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

$$5x+2y=-6$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$5x=-2y-6$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-2y-6\right)$$

Whakareatia \frac{1}{5} ki te -2y-6.

$$x=-\frac{2}{5}y-\frac{6}{5}$$

Whakakapia te \frac{-2y-6}{5} mō te x ki tērā atu whārite, 2x+5y=8.

$$2\left(-\frac{2}{5}y-\frac{6}{5}\right)+5y=8$$

Whakareatia 2 ki te \frac{-2y-6}{5}.

$$-\frac{4}{5}y-\frac{12}{5}+5y=8$$

Tāpiri -\frac{4y}{5} ki te 5y.

$$\frac{21}{5}y-\frac{12}{5}=8$$

Me tāpiri \frac{12}{5} ki ngā taha e rua o te whārite.

$$\frac{21}{5}y=\frac{52}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{21}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{52}{21}$$

Whakaurua te \frac{52}{21} mō y ki x=-\frac{2}{5}y-\frac{6}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{5}\times \frac{52}{21}-\frac{6}{5}$$

Whakareatia -\frac{2}{5} ki te \frac{52}{21} 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.

$$x=-\frac{104}{105}-\frac{6}{5}$$

Tāpiri -\frac{6}{5} ki te -\frac{104}{105} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{46}{21}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{46}{21},y=\frac{52}{21}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$5x+2y=-6,2x+5y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&2\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&2\\2&5\end{matrix}\right))\left(\begin{matrix}5&2\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\2&5\end{matrix}\right))\left(\begin{matrix}-6\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\2&5\end{matrix}\right))\left(\begin{matrix}-6\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\2&5\end{matrix}\right))\left(\begin{matrix}-6\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5\times 5-2\times 2}&-\frac{2}{5\times 5-2\times 2}\\-\frac{2}{5\times 5-2\times 2}&\frac{5}{5\times 5-2\times 2}\end{matrix}\right)\left(\begin{matrix}-6\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}&-\frac{2}{21}\\-\frac{2}{21}&\frac{5}{21}\end{matrix}\right)\left(\begin{matrix}-6\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}\left(-6\right)-\frac{2}{21}\times 8\\-\frac{2}{21}\left(-6\right)+\frac{5}{21}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{46}{21}\\\frac{52}{21}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{46}{21},y=\frac{52}{21}$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$5x+2y=-6,2x+5y=8$$

Kia ōrite ai a 5x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$2\times 5x+2\times 2y=2\left(-6\right),5\times 2x+5\times 5y=5\times 8$$

Whakarūnātia.

$$10x+4y=-12,10x+25y=40$$

Me tango 10x+25y=40 mai i 10x+4y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+4y-25y=-12-40$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-25y=-12-40$$

Tāpiri 4y ki te -25y.

$$-21y=-12-40$$

Tāpiri -12 ki te -40.

$$-21y=-52$$

Whakawehea ngā taha e rua ki te -21.

$$y=\frac{52}{21}$$

Whakaurua te \frac{52}{21} mō y ki 2x+5y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+5\times \frac{52}{21}=8$$

Whakareatia 5 ki te \frac{52}{21}.

$$2x+\frac{260}{21}=8$$

Me tango \frac{260}{21} mai i ngā taha e rua o te whārite.

$$2x=-\frac{92}{21}$$

Whakawehea ngā taha e rua ki te 2.

$$x=-\frac{46}{21}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{46}{21},y=\frac{52}{21}$$