Whakaoti i te {l}{2x+5y=11}{x=2y+1}

$$\left. \begin{array}{l}{ 2 x + 5 y = 11 }\\{ x = 2 y + 1 }\end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-2y=1$$

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.

$$2x+5y=11,x-2y=1$$

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

$$2x+5y=11$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$2x=-5y+11$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y+11\right)$$

Whakareatia \frac{1}{2} ki te -5y+11.

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

Whakakapia te \frac{-5y+11}{2} mō te x ki tērā atu whārite, x-2y=1.

$$-\frac{5}{2}y+\frac{11}{2}-2y=1$$

Tāpiri -\frac{5y}{2} ki te -2y.

$$-\frac{9}{2}y+\frac{11}{2}=1$$

Me tango \frac{11}{2} mai i ngā taha e rua o te whārite.

$$-\frac{9}{2}y=-\frac{9}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=-\frac{5}{2}y+\frac{11}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Tāpiri \frac{11}{2} ki te -\frac{5}{2} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-2y=1$$

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

$$2x+5y=11,x-2y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\1&-2\end{matrix}\right))\left(\begin{matrix}11\\1\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}&\frac{5}{9}\\\frac{1}{9}&-\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}11\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}\times 11+\frac{5}{9}\\\frac{1}{9}\times 11-\frac{2}{9}\end{matrix}\right)$$

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 2y mai i ngā taha e rua.

$$x-2y=1$$

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.

$$2x+5y=11,x-2y=1$$

Kia ōrite ai a 2x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$2x+5y=11,2x+2\left(-2\right)y=2$$

Whakarūnātia.

$$2x+5y=11,2x-4y=2$$

Me tango 2x-4y=2 mai i 2x+5y=11 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+5y+4y=11-2$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y+4y=11-2$$

Tāpiri 5y ki te 4y.

$$9y=11-2$$

Tāpiri 11 ki te -2.

$$9y=9$$

Whakawehea ngā taha e rua ki te 9.

$$y=1$$

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

$$x-2=1$$

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

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$