Whakaoti i te {l}{6x+7y=-4}{6x+4y=8}

$$\left. \begin{array} { l } { 6 x + 7 y = - 4 } \\ { 6 x + 4 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=-4$$

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.

$$6x+7y=-4,6x+4y=8$$

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

$$6x+7y=-4$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$6x=-7y-4$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-7y-4\right)$$

Whakareatia \frac{1}{6} ki te -7y-4.

$$x=-\frac{7}{6}y-\frac{2}{3}$$

Whakakapia te -\frac{7y}{6}-\frac{2}{3} mō te x ki tērā atu whārite, 6x+4y=8.

$$6\left(-\frac{7}{6}y-\frac{2}{3}\right)+4y=8$$

Whakareatia 6 ki te -\frac{7y}{6}-\frac{2}{3}.

$$-7y-4+4y=8$$

Tāpiri -7y ki te 4y.

$$-3y-4=8$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$-3y=12$$

Whakawehea ngā taha e rua ki te -3.

$$y=-4$$

Whakaurua te -4 mō y ki x=-\frac{7}{6}y-\frac{2}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{6}\left(-4\right)-\frac{2}{3}$$

Whakareatia -\frac{7}{6} ki te -4.

$$x=\frac{14-2}{3}$$

Tāpiri -\frac{2}{3} ki te \frac{14}{3} 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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-4$$

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.

$$6x+7y=-4,6x+4y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&7\\6&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}6&7\\6&4\end{matrix}\right))\left(\begin{matrix}6&7\\6&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\6&4\end{matrix}\right))\left(\begin{matrix}-4\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\6&4\end{matrix}\right))\left(\begin{matrix}-4\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\6&4\end{matrix}\right))\left(\begin{matrix}-4\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{6\times 4-7\times 6}&-\frac{7}{6\times 4-7\times 6}\\-\frac{6}{6\times 4-7\times 6}&\frac{6}{6\times 4-7\times 6}\end{matrix}\right)\left(\begin{matrix}-4\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{9}&\frac{7}{18}\\\frac{1}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}-4\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{9}\left(-4\right)+\frac{7}{18}\times 8\\\frac{1}{3}\left(-4\right)-\frac{1}{3}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=-4$$

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.

$$6x+7y=-4,6x+4y=8$$

Me tango 6x+4y=8 mai i 6x+7y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+7y-4y=-4-8$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7y-4y=-4-8$$

Tāpiri 7y ki te -4y.

$$3y=-4-8$$

Tāpiri -4 ki te -8.

$$3y=-12$$

Whakawehea ngā taha e rua ki te 3.

$$y=-4$$

Whakaurua te -4 mō y ki 6x+4y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+4\left(-4\right)=8$$

Whakareatia 4 ki te -4.

$$6x-16=8$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$6x=24$$

Whakawehea ngā taha e rua ki te 6.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-4$$