Whakaoti i te {l}{2x+5y=20}{3x-2y=11}

$$\left. \begin{array} { l } { 2 x + 5 y = 20 } \\ { 3 x - 2 y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=2$$

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.

$$2x+5y=20,3x-2y=11$$

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

$$2x+5y=20$$

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

$$2x=-5y+20$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y+20\right)$$

Whakareatia \frac{1}{2} ki te -5y+20.

$$x=-\frac{5}{2}y+10$$

Whakakapia te -\frac{5y}{2}+10 mō te x ki tērā atu whārite, 3x-2y=11.

$$3\left(-\frac{5}{2}y+10\right)-2y=11$$

Whakareatia 3 ki te -\frac{5y}{2}+10.

$$-\frac{15}{2}y+30-2y=11$$

Tāpiri -\frac{15y}{2} ki te -2y.

$$-\frac{19}{2}y+30=11$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$-\frac{19}{2}y=-19$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{19}{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{5}{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.

$$x=-\frac{5}{2}\times 2+10$$

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

$$x=-5+10$$

Tāpiri 10 ki te -5.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=2$$

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.

$$2x+5y=20,3x-2y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&5\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&5\\3&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\3&-2\end{matrix}\right))\left(\begin{matrix}2&5\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\3&-2\end{matrix}\right))\left(\begin{matrix}20\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\3&-2\end{matrix}\right))\left(\begin{matrix}20\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\3&-2\end{matrix}\right))\left(\begin{matrix}20\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{2\left(-2\right)-5\times 3}&-\frac{5}{2\left(-2\right)-5\times 3}\\-\frac{3}{2\left(-2\right)-5\times 3}&\frac{2}{2\left(-2\right)-5\times 3}\end{matrix}\right)\left(\begin{matrix}20\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{19}&\frac{5}{19}\\\frac{3}{19}&-\frac{2}{19}\end{matrix}\right)\left(\begin{matrix}20\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{19}\times 20+\frac{5}{19}\times 11\\\frac{3}{19}\times 20-\frac{2}{19}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=5,y=2$$

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.

$$2x+5y=20,3x-2y=11$$

Kia ōrite ai a 2x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$3\times 2x+3\times 5y=3\times 20,2\times 3x+2\left(-2\right)y=2\times 11$$

Whakarūnātia.

$$6x+15y=60,6x-4y=22$$

Me tango 6x-4y=22 mai i 6x+15y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+15y+4y=60-22$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y+4y=60-22$$

Tāpiri 15y ki te 4y.

$$19y=60-22$$

Tāpiri 60 ki te -22.

$$19y=38$$

Whakawehea ngā taha e rua ki te 19.

$$y=2$$

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

$$3x-2\times 2=11$$

Whakareatia -2 ki te 2.

$$3x-4=11$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$3x=15$$

Whakawehea ngā taha e rua ki te 3.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=2$$