Whakaoti i te {l}{2x+y=7}{4x-y=5}

$$\left. \begin{array} { l } { 2 x + y = 7 } \\ { 4 x - y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=3$$

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.

$$2x+y=7,4x-y=5$$

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

$$2x+y=7$$

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

$$2x=-y+7$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y+7\right)$$

Whakareatia \frac{1}{2} ki te -y+7.

$$x=-\frac{1}{2}y+\frac{7}{2}$$

Whakakapia te \frac{-y+7}{2} mō te x ki tērā atu whārite, 4x-y=5.

$$4\left(-\frac{1}{2}y+\frac{7}{2}\right)-y=5$$

Whakareatia 4 ki te \frac{-y+7}{2}.

$$-2y+14-y=5$$

Tāpiri -2y ki te -y.

$$-3y+14=5$$

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

$$-3y=-9$$

Whakawehea ngā taha e rua ki te -3.

$$y=3$$

Whakaurua te 3 mō y ki x=-\frac{1}{2}y+\frac{7}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 3+\frac{7}{2}$$

Whakareatia -\frac{1}{2} ki te 3.

$$x=\frac{-3+7}{2}$$

Tāpiri \frac{7}{2} ki te -\frac{3}{2} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$

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.

$$2x+y=7,4x-y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&1\\4&-1\end{matrix}\right))\left(\begin{matrix}2&1\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\4&-1\end{matrix}\right))\left(\begin{matrix}7\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\4&-1\end{matrix}\right))\left(\begin{matrix}7\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\4&-1\end{matrix}\right))\left(\begin{matrix}7\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-4}&-\frac{1}{2\left(-1\right)-4}\\-\frac{4}{2\left(-1\right)-4}&\frac{2}{2\left(-1\right)-4}\end{matrix}\right)\left(\begin{matrix}7\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\\frac{2}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}7\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 7+\frac{1}{6}\times 5\\\frac{2}{3}\times 7-\frac{1}{3}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=3$$

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.

$$2x+y=7,4x-y=5$$

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.

$$4\times 2x+4y=4\times 7,2\times 4x+2\left(-1\right)y=2\times 5$$

Whakarūnātia.

$$8x+4y=28,8x-2y=10$$

Me tango 8x-2y=10 mai i 8x+4y=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+4y+2y=28-10$$

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.

$$4y+2y=28-10$$

Tāpiri 4y ki te 2y.

$$6y=28-10$$

Tāpiri 28 ki te -10.

$$6y=18$$

Whakawehea ngā taha e rua ki te 6.

$$y=3$$

Whakaurua te 3 mō y ki 4x-y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-3=5$$

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

$$4x=8$$

Whakawehea ngā taha e rua ki te 4.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$