Whakaoti i te {l}{2x+3y=20}{7x+2y=53}

$$\left. \begin{array} { l } { 2 x + 3 y = 20 } \\ { 7 x + 2 y = 53 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=2$$

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.

$$2x+3y=20,7x+2y=53$$

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

$$2x+3y=20$$

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

$$2x=-3y+20$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+20\right)$$

Whakareatia \frac{1}{2} ki te -3y+20.

$$x=-\frac{3}{2}y+10$$

Whakakapia te -\frac{3y}{2}+10 mō te x ki tērā atu whārite, 7x+2y=53.

$$7\left(-\frac{3}{2}y+10\right)+2y=53$$

Whakareatia 7 ki te -\frac{3y}{2}+10.

$$-\frac{21}{2}y+70+2y=53$$

Tāpiri -\frac{21y}{2} ki te 2y.

$$-\frac{17}{2}y+70=53$$

Me tango 70 mai i ngā taha e rua o te whārite.

$$-\frac{17}{2}y=-17$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{17}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{3}{2}y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 2+10$$

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

$$x=-3+10$$

Tāpiri 10 ki te -3.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=2$$

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.

$$2x+3y=20,7x+2y=53$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\53\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\7&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\7&2\end{matrix}\right))\left(\begin{matrix}2&3\\7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\7&2\end{matrix}\right))\left(\begin{matrix}20\\53\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\7&2\end{matrix}\right))\left(\begin{matrix}20\\53\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\7&2\end{matrix}\right))\left(\begin{matrix}20\\53\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-3\times 7}&-\frac{3}{2\times 2-3\times 7}\\-\frac{7}{2\times 2-3\times 7}&\frac{2}{2\times 2-3\times 7}\end{matrix}\right)\left(\begin{matrix}20\\53\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{17}&\frac{3}{17}\\\frac{7}{17}&-\frac{2}{17}\end{matrix}\right)\left(\begin{matrix}20\\53\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{17}\times 20+\frac{3}{17}\times 53\\\frac{7}{17}\times 20-\frac{2}{17}\times 53\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=2$$

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.

$$2x+3y=20,7x+2y=53$$

Kia ōrite ai a 2x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$7\times 2x+7\times 3y=7\times 20,2\times 7x+2\times 2y=2\times 53$$

Whakarūnātia.

$$14x+21y=140,14x+4y=106$$

Me tango 14x+4y=106 mai i 14x+21y=140 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$14x-14x+21y-4y=140-106$$

Tāpiri 14x ki te -14x. Ka whakakore atu ngā kupu 14x me -14x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$21y-4y=140-106$$

Tāpiri 21y ki te -4y.

$$17y=140-106$$

Tāpiri 140 ki te -106.

$$17y=34$$

Whakawehea ngā taha e rua ki te 17.

$$y=2$$

Whakaurua te 2 mō y ki 7x+2y=53. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+2\times 2=53$$

Whakareatia 2 ki te 2.

$$7x+4=53$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$7x=49$$

Whakawehea ngā taha e rua ki te 7.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=2$$