Whakaoti i te {l}{2x+y=-19}{x+4y=11}

$$\left. \begin{array} { l } { 2 x + y = - 19 } \\ { x + 4 y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{87}{7} = -12\frac{3}{7} \approx -12.428571429$$

$$y = \frac{41}{7} = 5\frac{6}{7} \approx 5.857142857$$

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.

$$2x+y=-19,x+4y=11$$

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

$$2x+y=-19$$

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

$$2x=-y-19$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y-19\right)$$

Whakareatia \frac{1}{2} ki te -y-19.

$$x=-\frac{1}{2}y-\frac{19}{2}$$

Whakakapia te \frac{-y-19}{2} mō te x ki tērā atu whārite, x+4y=11.

$$-\frac{1}{2}y-\frac{19}{2}+4y=11$$

Tāpiri -\frac{y}{2} ki te 4y.

$$\frac{7}{2}y-\frac{19}{2}=11$$

Me tāpiri \frac{19}{2} ki ngā taha e rua o te whārite.

$$\frac{7}{2}y=\frac{41}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{41}{7}$$

Whakaurua te \frac{41}{7} mō y ki x=-\frac{1}{2}y-\frac{19}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times \frac{41}{7}-\frac{19}{2}$$

Whakareatia -\frac{1}{2} ki te \frac{41}{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.

$$x=-\frac{41}{14}-\frac{19}{2}$$

Tāpiri -\frac{19}{2} ki te -\frac{41}{14} 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.

$$x=-\frac{87}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{87}{7},y=\frac{41}{7}$$

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.

$$2x+y=-19,x+4y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\1&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-19\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\1&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\1&4\end{matrix}\right))\left(\begin{matrix}2&1\\1&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&4\end{matrix}\right))\left(\begin{matrix}-19\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&4\end{matrix}\right))\left(\begin{matrix}-19\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&4\end{matrix}\right))\left(\begin{matrix}-19\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{2\times 4-1}&-\frac{1}{2\times 4-1}\\-\frac{1}{2\times 4-1}&\frac{2}{2\times 4-1}\end{matrix}\right)\left(\begin{matrix}-19\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7}&-\frac{1}{7}\\-\frac{1}{7}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}-19\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7}\left(-19\right)-\frac{1}{7}\times 11\\-\frac{1}{7}\left(-19\right)+\frac{2}{7}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{87}{7}\\\frac{41}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{87}{7},y=\frac{41}{7}$$

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.

$$2x+y=-19,x+4y=11$$

Kia ōrite ai a 2x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$2x+y=-19,2x+2\times 4y=2\times 11$$

Whakarūnātia.

$$2x+y=-19,2x+8y=22$$

Me tango 2x+8y=22 mai i 2x+y=-19 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+y-8y=-19-22$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-8y=-19-22$$

Tāpiri y ki te -8y.

$$-7y=-19-22$$

Tāpiri -19 ki te -22.

$$-7y=-41$$

Whakawehea ngā taha e rua ki te -7.

$$y=\frac{41}{7}$$

Whakaurua te \frac{41}{7} mō y ki x+4y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+4\times \frac{41}{7}=11$$

Whakareatia 4 ki te \frac{41}{7}.

$$x+\frac{164}{7}=11$$

Me tango \frac{164}{7} mai i ngā taha e rua o te whārite.

$$x=-\frac{87}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{87}{7},y=\frac{41}{7}$$