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

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

Whakaoti mō x, y

$$x=3$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 7y mai i ngā taha e rua.

$$4x-7y=5$$

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.

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

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

$$x-y=2$$

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

$$x=y+2$$

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

$$4\left(y+2\right)-7y=5$$

Whakareatia 4 ki te y+2.

$$4y+8-7y=5$$

Tāpiri 4y ki te -7y.

$$-3y+8=5$$

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

$$-3y=-3$$

Whakawehea ngā taha e rua ki te -3.

$$y=1$$

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

$$x=1+2$$

Tāpiri 2 ki te 1.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 7y mai i ngā taha e rua.

$$4x-7y=5$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\4&-7\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{3}&-\frac{1}{3}\\\frac{4}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}2\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{3}\times 2-\frac{1}{3}\times 5\\\frac{4}{3}\times 2-\frac{1}{3}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 7y mai i ngā taha e rua.

$$4x-7y=5$$

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.

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

Kia ōrite ai a x 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 1.

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

Whakarūnātia.

$$4x-4y=8,4x-7y=5$$

Me tango 4x-7y=5 mai i 4x-4y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x-4y+7y=8-5$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y+7y=8-5$$

Tāpiri -4y ki te 7y.

$$3y=8-5$$

Tāpiri 8 ki te -5.

$$3y=3$$

Whakawehea ngā taha e rua ki te 3.

$$y=1$$

Whakaurua te 1 mō y ki 4x-7y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-7=5$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$4x=12$$

Whakawehea ngā taha e rua ki te 4.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$