Whakaoti i te {l}{2x+14y=-28}{-4x-14y=28}

$$\left\{ \begin{array} { l } { 2 x + 14 y = - 28 } \\ { - 4 x - 14 y = 28 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=-2$$

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.

$$2x+14y=-28,-4x-14y=28$$

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

$$2x+14y=-28$$

Me tango 14y mai i ngā taha e rua o te whārite.

$$2x=-14y-28$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-14y-28\right)$$

Whakareatia \frac{1}{2} ki te -14y-28.

$$x=-7y-14$$

Whakakapia te -7y-14 mō te x ki tērā atu whārite, -4x-14y=28.

$$-4\left(-7y-14\right)-14y=28$$

Whakareatia -4 ki te -7y-14.

$$28y+56-14y=28$$

Tāpiri 28y ki te -14y.

$$14y+56=28$$

Me tango 56 mai i ngā taha e rua o te whārite.

$$14y=-28$$

Whakawehea ngā taha e rua ki te 14.

$$y=-2$$

Whakaurua te -2 mō y ki x=-7y-14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-7\left(-2\right)-14$$

Whakareatia -7 ki te -2.

$$x=14-14$$

Tāpiri -14 ki te 14.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$

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.

$$2x+14y=-28,-4x-14y=28$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&14\\-4&-14\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-28\\28\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&14\\-4&-14\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&14\\-4&-14\end{matrix}\right))\left(\begin{matrix}2&14\\-4&-14\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&14\\-4&-14\end{matrix}\right))\left(\begin{matrix}-28\\28\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&14\\-4&-14\end{matrix}\right))\left(\begin{matrix}-28\\28\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&14\\-4&-14\end{matrix}\right))\left(\begin{matrix}-28\\28\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{14}{2\left(-14\right)-14\left(-4\right)}&-\frac{14}{2\left(-14\right)-14\left(-4\right)}\\-\frac{-4}{2\left(-14\right)-14\left(-4\right)}&\frac{2}{2\left(-14\right)-14\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}-28\\28\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&-\frac{1}{2}\\\frac{1}{7}&\frac{1}{14}\end{matrix}\right)\left(\begin{matrix}-28\\28\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\left(-28\right)-\frac{1}{2}\times 28\\\frac{1}{7}\left(-28\right)+\frac{1}{14}\times 28\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-2$$

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.

$$2x+14y=-28,-4x-14y=28$$

Kia ōrite ai a 2x me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-4\times 2x-4\times 14y=-4\left(-28\right),2\left(-4\right)x+2\left(-14\right)y=2\times 28$$

Whakarūnātia.

$$-8x-56y=112,-8x-28y=56$$

Me tango -8x-28y=56 mai i -8x-56y=112 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x-56y+28y=112-56$$

Tāpiri -8x ki te 8x. Ka whakakore atu ngā kupu -8x me 8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-56y+28y=112-56$$

Tāpiri -56y ki te 28y.

$$-28y=112-56$$

Tāpiri 112 ki te -56.

$$-28y=56$$

Whakawehea ngā taha e rua ki te -28.

$$y=-2$$

Whakaurua te -2 mō y ki -4x-14y=28. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x-14\left(-2\right)=28$$

Whakareatia -14 ki te -2.

$$-4x+28=28$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$-4x=0$$

Whakawehea ngā taha e rua ki te -4.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$