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

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

Whakaoti mō x, y

$$x=0$$

$$y=-1$$

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.

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

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

$$-5x-8y=8$$

Me tāpiri 8y ki ngā taha e rua o te whārite.

$$-5x=8y+8$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{1}{5}\left(8y+8\right)$$

Whakareatia -\frac{1}{5} ki te 8+8y.

$$x=-\frac{8}{5}y-\frac{8}{5}$$

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

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

Whakareatia -5 ki te \frac{-8y-8}{5}.

$$8y+8+6y=-6$$

Tāpiri 8y ki te 6y.

$$14y+8=-6$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$14y=-14$$

Whakawehea ngā taha e rua ki te 14.

$$y=-1$$

Whakaurua te -1 mō y ki x=-\frac{8}{5}y-\frac{8}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{8}{5}\left(-1\right)-\frac{8}{5}$$

Whakareatia -\frac{8}{5} ki te -1.

$$x=\frac{8-8}{5}$$

Tāpiri -\frac{8}{5} ki te \frac{8}{5} 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.

$$x=0,y=-1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&-8\\-5&6\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{35}&-\frac{4}{35}\\-\frac{1}{14}&\frac{1}{14}\end{matrix}\right)\left(\begin{matrix}8\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{35}\times 8-\frac{4}{35}\left(-6\right)\\-\frac{1}{14}\times 8+\frac{1}{14}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-1$$

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.

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

Me tango -5x+6y=-6 mai i -5x-8y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

Tāpiri -5x ki te 5x. Ka whakakore atu ngā kupu -5x me 5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y-6y=8+6$$

Tāpiri -8y ki te -6y.

$$-14y=8+6$$

Tāpiri 8 ki te 6.

$$-14y=14$$

Whakawehea ngā taha e rua ki te -14.

$$y=-1$$

Whakaurua te -1 mō y ki -5x+6y=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+6\left(-1\right)=-6$$

Whakareatia 6 ki te -1.

$$-5x-6=-6$$

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

$$-5x=0$$

Whakawehea ngā taha e rua ki te -5.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-1$$