Whakaoti i te {l}{x+2y=3}{5x-y=10}

$$\left. \begin{array} { l } { x + 2 y = 3 } \\ { 5 x - y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{23}{11} = 2\frac{1}{11} \approx 2.090909091$$

$$y=\frac{5}{11}\approx 0.454545455$$

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.

$$x+2y=3,5x-y=10$$

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

$$x+2y=3$$

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

$$x=-2y+3$$

Whakakapia te -2y+3 mō te x ki tērā atu whārite, 5x-y=10.

$$5\left(-2y+3\right)-y=10$$

Whakareatia 5 ki te -2y+3.

$$-10y+15-y=10$$

Tāpiri -10y ki te -y.

$$-11y+15=10$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-11y=-5$$

Whakawehea ngā taha e rua ki te -11.

$$y=\frac{5}{11}$$

Whakaurua te \frac{5}{11} mō y ki x=-2y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{5}{11}+3$$

Whakareatia -2 ki te \frac{5}{11}.

$$x=-\frac{10}{11}+3$$

Tāpiri 3 ki te -\frac{10}{11}.

$$x=\frac{23}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{23}{11},y=\frac{5}{11}$$

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.

$$x+2y=3,5x-y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\5&-1\end{matrix}\right))\left(\begin{matrix}3\\10\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{2}{11}\\\frac{5}{11}&-\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}3\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 3+\frac{2}{11}\times 10\\\frac{5}{11}\times 3-\frac{1}{11}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23}{11}\\\frac{5}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{23}{11},y=\frac{5}{11}$$

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.

$$x+2y=3,5x-y=10$$

Kia ōrite ai a x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$5x+5\times 2y=5\times 3,5x-y=10$$

Whakarūnātia.

$$5x+10y=15,5x-y=10$$

Me tango 5x-y=10 mai i 5x+10y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x+10y+y=15-10$$

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y+y=15-10$$

Tāpiri 10y ki te y.

$$11y=15-10$$

Tāpiri 15 ki te -10.

$$11y=5$$

Whakawehea ngā taha e rua ki te 11.

$$y=\frac{5}{11}$$

Whakaurua te \frac{5}{11} mō y ki 5x-y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-\frac{5}{11}=10$$

Me tāpiri \frac{5}{11} ki ngā taha e rua o te whārite.

$$5x=\frac{115}{11}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{23}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{23}{11},y=\frac{5}{11}$$