Whakaoti i te {l}{2x-3y=4}{4x+y=-6}

$$\left\{ \begin{array} { l } { 2 x - 3 y = 4 } \\ { 4 x + y = - 6 } \end{array} \right.$$

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.

$$2x-3y=4,4x+y=-6$$

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

$$2x-3y=4$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$2x=3y+4$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y+4\right)$$

Whakareatia \frac{1}{2} ki te 3y+4.

$$x=\frac{3}{2}y+2$$

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

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

Whakareatia 4 ki te \frac{3y}{2}+2.

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

Tāpiri 6y ki te y.

$$7y+8=-6$$

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

$$7y=-14$$

Whakawehea ngā taha e rua ki te 7.

$$y=-2$$

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

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

Whakareatia \frac{3}{2} ki te -2.

$$x=-3+2$$

Tāpiri 2 ki te -3.

$$x=-1$$

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.

$$2x-3y=4,4x+y=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\4&1\end{matrix}\right))\left(\begin{matrix}2&-3\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&1\end{matrix}\right))\left(\begin{matrix}4\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&1\end{matrix}\right))\left(\begin{matrix}4\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&1\end{matrix}\right))\left(\begin{matrix}4\\-6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-3\times 4\right)}&-\frac{-3}{2-\left(-3\times 4\right)}\\-\frac{4}{2-\left(-3\times 4\right)}&\frac{2}{2-\left(-3\times 4\right)}\end{matrix}\right)\left(\begin{matrix}4\\-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}&\frac{3}{14}\\-\frac{2}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}4\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}\times 4+\frac{3}{14}\left(-6\right)\\-\frac{2}{7}\times 4+\frac{1}{7}\left(-6\right)\end{matrix}\right)$$

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.

$$2x-3y=4,4x+y=-6$$

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.

$$4\times 2x+4\left(-3\right)y=4\times 4,2\times 4x+2y=2\left(-6\right)$$

Whakarūnātia.

$$8x-12y=16,8x+2y=-12$$

Me tango 8x+2y=-12 mai i 8x-12y=16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-12y-2y=16+12$$

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.

$$-12y-2y=16+12$$

Tāpiri -12y ki te -2y.

$$-14y=16+12$$

Tāpiri 16 ki te 12.

$$-14y=28$$

Whakawehea ngā taha e rua ki te -14.

$$y=-2$$

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

$$4x-2=-6$$

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

$$4x=-4$$

Whakawehea ngā taha e rua ki te 4.

$$x=-1$$

Kua oti te pūnaha te whakatau.

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