Whakaoti i te {l}{5x+2y=10}{4x+y=8}

$$\left. \begin{array} { l } { 5 x + 2 y = 10 } \\ { 4 x + y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=0$$

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.

$$5x+2y=10,4x+y=8$$

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

$$5x+2y=10$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$5x=-2y+10$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-2y+10\right)$$

Whakareatia \frac{1}{5} ki te -2y+10.

$$x=-\frac{2}{5}y+2$$

Whakakapia te -\frac{2y}{5}+2 mō te x ki tērā atu whārite, 4x+y=8.

$$4\left(-\frac{2}{5}y+2\right)+y=8$$

Whakareatia 4 ki te -\frac{2y}{5}+2.

$$-\frac{8}{5}y+8+y=8$$

Tāpiri -\frac{8y}{5} ki te y.

$$-\frac{3}{5}y+8=8$$

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

$$-\frac{3}{5}y=0$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{2}{5}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=0$$

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.

$$5x+2y=10,4x+y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&2\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&2\\4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&2\\4&1\end{matrix}\right))\left(\begin{matrix}5&2\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\4&1\end{matrix}\right))\left(\begin{matrix}10\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\4&1\end{matrix}\right))\left(\begin{matrix}10\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\4&1\end{matrix}\right))\left(\begin{matrix}10\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5-2\times 4}&-\frac{2}{5-2\times 4}\\-\frac{4}{5-2\times 4}&\frac{5}{5-2\times 4}\end{matrix}\right)\left(\begin{matrix}10\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{2}{3}\\\frac{4}{3}&-\frac{5}{3}\end{matrix}\right)\left(\begin{matrix}10\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 10+\frac{2}{3}\times 8\\\frac{4}{3}\times 10-\frac{5}{3}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=0$$

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.

$$5x+2y=10,4x+y=8$$

Kia ōrite ai a 5x 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 5.

$$4\times 5x+4\times 2y=4\times 10,5\times 4x+5y=5\times 8$$

Whakarūnātia.

$$20x+8y=40,20x+5y=40$$

Me tango 20x+5y=40 mai i 20x+8y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x+8y-5y=40-40$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-5y=40-40$$

Tāpiri 8y ki te -5y.

$$3y=40-40$$

Tāpiri 40 ki te -40.

$$3y=0$$

Whakawehea ngā taha e rua ki te 3.

$$y=0$$

Whakaurua te 0 mō y ki 4x+y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x=8$$

Whakawehea ngā taha e rua ki te 4.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=0$$