Whakaoti i te {l}{7x-6y=-30}{x-4y=-20}

$$\left\{ \begin{array} { l } { 7 x - 6 y = - 30 } \\ { x - 4 y = - 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=5$$

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.

$$7x-6y=-30,x-4y=-20$$

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

$$7x-6y=-30$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$7x=6y-30$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(6y-30\right)$$

Whakareatia \frac{1}{7} ki te -30+6y.

$$x=\frac{6}{7}y-\frac{30}{7}$$

Whakakapia te \frac{-30+6y}{7} mō te x ki tērā atu whārite, x-4y=-20.

$$\frac{6}{7}y-\frac{30}{7}-4y=-20$$

Tāpiri \frac{6y}{7} ki te -4y.

$$-\frac{22}{7}y-\frac{30}{7}=-20$$

Me tāpiri \frac{30}{7} ki ngā taha e rua o te whārite.

$$-\frac{22}{7}y=-\frac{110}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{22}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

Whakaurua te 5 mō y ki x=\frac{6}{7}y-\frac{30}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{6}{7}\times 5-\frac{30}{7}$$

Whakareatia \frac{6}{7} ki te 5.

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

Tāpiri -\frac{30}{7} ki te \frac{30}{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.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=5$$

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.

$$7x-6y=-30,x-4y=-20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-6\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-30\\-20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-6\\1&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-6\\1&-4\end{matrix}\right))\left(\begin{matrix}7&-6\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-6\\1&-4\end{matrix}\right))\left(\begin{matrix}-30\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-6\\1&-4\end{matrix}\right))\left(\begin{matrix}-30\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-6\\1&-4\end{matrix}\right))\left(\begin{matrix}-30\\-20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7\left(-4\right)-\left(-6\right)}&-\frac{-6}{7\left(-4\right)-\left(-6\right)}\\-\frac{1}{7\left(-4\right)-\left(-6\right)}&\frac{7}{7\left(-4\right)-\left(-6\right)}\end{matrix}\right)\left(\begin{matrix}-30\\-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}&-\frac{3}{11}\\\frac{1}{22}&-\frac{7}{22}\end{matrix}\right)\left(\begin{matrix}-30\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\left(-30\right)-\frac{3}{11}\left(-20\right)\\\frac{1}{22}\left(-30\right)-\frac{7}{22}\left(-20\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=5$$

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.

$$7x-6y=-30,x-4y=-20$$

Kia ōrite ai a 7x 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 7.

$$7x-6y=-30,7x+7\left(-4\right)y=7\left(-20\right)$$

Whakarūnātia.

$$7x-6y=-30,7x-28y=-140$$

Me tango 7x-28y=-140 mai i 7x-6y=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-7x-6y+28y=-30+140$$

Tāpiri 7x ki te -7x. Ka whakakore atu ngā kupu 7x me -7x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y+28y=-30+140$$

Tāpiri -6y ki te 28y.

$$22y=-30+140$$

Tāpiri -30 ki te 140.

$$22y=110$$

Whakawehea ngā taha e rua ki te 22.

$$y=5$$

Whakaurua te 5 mō y ki x-4y=-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-4\times 5=-20$$

Whakareatia -4 ki te 5.

$$x-20=-20$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=5$$