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

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

Whakaoti mō x, y

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$y=-1$$

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.

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

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

$$2x-y=4$$

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

$$2x=y+4$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

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

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

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

$$2y+8+3y=3$$

Tāpiri 2y ki te 3y.

$$5y+8=3$$

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

$$5y=-5$$

Whakawehea ngā taha e rua ki te 5.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{1}{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.

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

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

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

Tāpiri 2 ki te -\frac{1}{2}.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{2},y=-1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}&\frac{1}{10}\\-\frac{2}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}4\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}\times 4+\frac{1}{10}\times 3\\-\frac{2}{5}\times 4+\frac{1}{5}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3}{2},y=-1$$

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.

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

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.

$$4\times 2x+4\left(-1\right)y=4\times 4,2\times 4x+2\times 3y=2\times 3$$

Whakarūnātia.

$$8x-4y=16,8x+6y=6$$

Me tango 8x+6y=6 mai i 8x-4y=16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-4y-6y=16-6$$

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.

$$-4y-6y=16-6$$

Tāpiri -4y ki te -6y.

$$-10y=16-6$$

Tāpiri 16 ki te -6.

$$-10y=10$$

Whakawehea ngā taha e rua ki te -10.

$$y=-1$$

Whakaurua te -1 mō y ki 4x+3y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+3\left(-1\right)=3$$

Whakareatia 3 ki te -1.

$$4x-3=3$$

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

$$4x=6$$

Whakawehea ngā taha e rua ki te 4.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{2},y=-1$$