Whakaoti i te {l}{8+4x=2y}{-4x=14-3y}

$$\left. \begin{array} { l } { 8 + 4 x = 2 y } \\ { - 4 x = 14 - 3 y } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 2y mai i ngā taha e rua.

$$8+4x-2y=0$$

Tangohia te 8 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$4x-2y=-8$$

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

$$-4x+3y=14$$

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.

$$4x-2y=-8,-4x+3y=14$$

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

$$4x-2y=-8$$

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

$$4x=2y-8$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(2y-8\right)$$

Whakareatia \frac{1}{4} ki te -8+2y.

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

Whakakapia te \frac{y}{2}-2 mō te x ki tērā atu whārite, -4x+3y=14.

$$-4\left(\frac{1}{2}y-2\right)+3y=14$$

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

$$-2y+8+3y=14$$

Tāpiri -2y ki te 3y.

$$y+8=14$$

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

$$y=6$$

Whakaurua te 6 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.

$$x=\frac{1}{2}\times 6-2$$

Whakareatia \frac{1}{2} ki te 6.

$$x=3-2$$

Tāpiri -2 ki te 3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=6$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 2y mai i ngā taha e rua.

$$8+4x-2y=0$$

Tangohia te 8 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$4x-2y=-8$$

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

$$-4x+3y=14$$

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

$$4x-2y=-8,-4x+3y=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-2\\-4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-8\\14\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&-2\\-4&3\end{matrix}\right))\left(\begin{matrix}4&-2\\-4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\-4&3\end{matrix}\right))\left(\begin{matrix}-8\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\-4&3\end{matrix}\right))\left(\begin{matrix}-8\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\-4&3\end{matrix}\right))\left(\begin{matrix}-8\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4\times 3-\left(-2\left(-4\right)\right)}&-\frac{-2}{4\times 3-\left(-2\left(-4\right)\right)}\\-\frac{-4}{4\times 3-\left(-2\left(-4\right)\right)}&\frac{4}{4\times 3-\left(-2\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}-8\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&\frac{1}{2}\\1&1\end{matrix}\right)\left(\begin{matrix}-8\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\left(-8\right)+\frac{1}{2}\times 14\\-8+14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=6$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 2y mai i ngā taha e rua.

$$8+4x-2y=0$$

Tangohia te 8 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$4x-2y=-8$$

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

$$-4x+3y=14$$

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.

$$4x-2y=-8,-4x+3y=14$$

Kia ōrite ai a 4x 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 4.

$$-4\times 4x-4\left(-2\right)y=-4\left(-8\right),4\left(-4\right)x+4\times 3y=4\times 14$$

Whakarūnātia.

$$-16x+8y=32,-16x+12y=56$$

Me tango -16x+12y=56 mai i -16x+8y=32 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-16x+16x+8y-12y=32-56$$

Tāpiri -16x ki te 16x. Ka whakakore atu ngā kupu -16x me 16x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-12y=32-56$$

Tāpiri 8y ki te -12y.

$$-4y=32-56$$

Tāpiri 32 ki te -56.

$$-4y=-24$$

Whakawehea ngā taha e rua ki te -4.

$$y=6$$

Whakaurua te 6 mō y ki -4x+3y=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+3\times 6=14$$

Whakareatia 3 ki te 6.

$$-4x+18=14$$

Me tango 18 mai i 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.

$$x=1,y=6$$