Whakaoti i te {l}{2x+3y=19}{4x+11y=53}

$$\left. \begin{array} { l } { 2 x + 3 y = 19 } \\ { 4 x + 11 y = 53 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=3$$

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.

$$2x+3y=19,4x+11y=53$$

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

$$2x+3y=19$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+19$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+19\right)$$

Whakareatia \frac{1}{2} ki te -3y+19.

$$x=-\frac{3}{2}y+\frac{19}{2}$$

Whakakapia te \frac{-3y+19}{2} mō te x ki tērā atu whārite, 4x+11y=53.

$$4\left(-\frac{3}{2}y+\frac{19}{2}\right)+11y=53$$

Whakareatia 4 ki te \frac{-3y+19}{2}.

$$-6y+38+11y=53$$

Tāpiri -6y ki te 11y.

$$5y+38=53$$

Me tango 38 mai i ngā taha e rua o te whārite.

$$5y=15$$

Whakawehea ngā taha e rua ki te 5.

$$y=3$$

Whakaurua te 3 mō y ki x=-\frac{3}{2}y+\frac{19}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 3+\frac{19}{2}$$

Whakareatia -\frac{3}{2} ki te 3.

$$x=\frac{-9+19}{2}$$

Tāpiri \frac{19}{2} ki te -\frac{9}{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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=3$$

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.

$$2x+3y=19,4x+11y=53$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\4&11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}19\\53\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\4&11\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\4&11\end{matrix}\right))\left(\begin{matrix}2&3\\4&11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&11\end{matrix}\right))\left(\begin{matrix}19\\53\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&11\end{matrix}\right))\left(\begin{matrix}19\\53\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&11\end{matrix}\right))\left(\begin{matrix}19\\53\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{2\times 11-3\times 4}&-\frac{3}{2\times 11-3\times 4}\\-\frac{4}{2\times 11-3\times 4}&\frac{2}{2\times 11-3\times 4}\end{matrix}\right)\left(\begin{matrix}19\\53\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{10}&-\frac{3}{10}\\-\frac{2}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}19\\53\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{10}\times 19-\frac{3}{10}\times 53\\-\frac{2}{5}\times 19+\frac{1}{5}\times 53\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=3$$

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.

$$2x+3y=19,4x+11y=53$$

Kia ōrite ai a 2x 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 2.

$$4\times 2x+4\times 3y=4\times 19,2\times 4x+2\times 11y=2\times 53$$

Whakarūnātia.

$$8x+12y=76,8x+22y=106$$

Me tango 8x+22y=106 mai i 8x+12y=76 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+12y-22y=76-106$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-22y=76-106$$

Tāpiri 12y ki te -22y.

$$-10y=76-106$$

Tāpiri 76 ki te -106.

$$-10y=-30$$

Whakawehea ngā taha e rua ki te -10.

$$y=3$$

Whakaurua te 3 mō y ki 4x+11y=53. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+11\times 3=53$$

Whakareatia 11 ki te 3.

$$4x+33=53$$

Me tango 33 mai i ngā taha e rua o te whārite.

$$4x=20$$

Whakawehea ngā taha e rua ki te 4.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=3$$