Whakaoti i te {l}{7x+5y=-3}{-9x+y=-11}

$$\left. \begin{array} { l } { 7 x + 5 y = - 3 } \\ { - 9 x + y = - 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-2$$

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.

$$7x+5y=-3,-9x+y=-11$$

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

$$7x+5y=-3$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$7x=-5y-3$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-5y-3\right)$$

Whakareatia \frac{1}{7} ki te -5y-3.

$$x=-\frac{5}{7}y-\frac{3}{7}$$

Whakakapia te \frac{-5y-3}{7} mō te x ki tērā atu whārite, -9x+y=-11.

$$-9\left(-\frac{5}{7}y-\frac{3}{7}\right)+y=-11$$

Whakareatia -9 ki te \frac{-5y-3}{7}.

$$\frac{45}{7}y+\frac{27}{7}+y=-11$$

Tāpiri \frac{45y}{7} ki te y.

$$\frac{52}{7}y+\frac{27}{7}=-11$$

Me tango \frac{27}{7} mai i ngā taha e rua o te whārite.

$$\frac{52}{7}y=-\frac{104}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{52}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

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

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

Whakareatia -\frac{5}{7} ki te -2.

$$x=\frac{10-3}{7}$$

Tāpiri -\frac{3}{7} ki te \frac{10}{7} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-2$$

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.

$$7x+5y=-3,-9x+y=-11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&5\\-9&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&5\\-9&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&5\\-9&1\end{matrix}\right))\left(\begin{matrix}7&5\\-9&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\-9&1\end{matrix}\right))\left(\begin{matrix}-3\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\-9&1\end{matrix}\right))\left(\begin{matrix}-3\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\-9&1\end{matrix}\right))\left(\begin{matrix}-3\\-11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7-5\left(-9\right)}&-\frac{5}{7-5\left(-9\right)}\\-\frac{-9}{7-5\left(-9\right)}&\frac{7}{7-5\left(-9\right)}\end{matrix}\right)\left(\begin{matrix}-3\\-11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{52}&-\frac{5}{52}\\\frac{9}{52}&\frac{7}{52}\end{matrix}\right)\left(\begin{matrix}-3\\-11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{52}\left(-3\right)-\frac{5}{52}\left(-11\right)\\\frac{9}{52}\left(-3\right)+\frac{7}{52}\left(-11\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-2$$

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.

$$7x+5y=-3,-9x+y=-11$$

Kia ōrite ai a 7x me -9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$-9\times 7x-9\times 5y=-9\left(-3\right),7\left(-9\right)x+7y=7\left(-11\right)$$

Whakarūnātia.

$$-63x-45y=27,-63x+7y=-77$$

Me tango -63x+7y=-77 mai i -63x-45y=27 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-63x+63x-45y-7y=27+77$$

Tāpiri -63x ki te 63x. Ka whakakore atu ngā kupu -63x me 63x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-45y-7y=27+77$$

Tāpiri -45y ki te -7y.

$$-52y=27+77$$

Tāpiri 27 ki te 77.

$$-52y=104$$

Whakawehea ngā taha e rua ki te -52.

$$y=-2$$

Whakaurua te -2 mō y ki -9x+y=-11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-9x-2=-11$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$-9x=-9$$

Whakawehea ngā taha e rua ki te -9.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-2$$