Whakaoti i te {l}{3x-5y=11}{x+3y=13}

$$\left\{ \begin{array} { l } { 3 x - 5 y = 11 } \\ { x + 3 y = 13 } \end{array} \right.$$

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.

$$3x-5y=11,x+3y=13$$

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

$$3x-5y=11$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$3x=5y+11$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(5y+11\right)$$

Whakareatia \frac{1}{3} ki te 5y+11.

$$x=\frac{5}{3}y+\frac{11}{3}$$

Whakakapia te \frac{5y+11}{3} mō te x ki tērā atu whārite, x+3y=13.

$$\frac{5}{3}y+\frac{11}{3}+3y=13$$

Tāpiri \frac{5y}{3} ki te 3y.

$$\frac{14}{3}y+\frac{11}{3}=13$$

Me tango \frac{11}{3} mai i ngā taha e rua o te whārite.

$$\frac{14}{3}y=\frac{28}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{14}{3}, 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}{3}y+\frac{11}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Whakareatia \frac{5}{3} ki te 2.

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

Tāpiri \frac{11}{3} ki te \frac{10}{3} 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.

$$x=7$$

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.

$$3x-5y=11,x+3y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-5\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-5\\1&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-5\\1&3\end{matrix}\right))\left(\begin{matrix}3&-5\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\1&3\end{matrix}\right))\left(\begin{matrix}11\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\1&3\end{matrix}\right))\left(\begin{matrix}11\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\1&3\end{matrix}\right))\left(\begin{matrix}11\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-\left(-5\right)}&-\frac{-5}{3\times 3-\left(-5\right)}\\-\frac{1}{3\times 3-\left(-5\right)}&\frac{3}{3\times 3-\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}11\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{14}&\frac{5}{14}\\-\frac{1}{14}&\frac{3}{14}\end{matrix}\right)\left(\begin{matrix}11\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{14}\times 11+\frac{5}{14}\times 13\\-\frac{1}{14}\times 11+\frac{3}{14}\times 13\end{matrix}\right)$$

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.

$$3x-5y=11,x+3y=13$$

Kia ōrite ai a 3x 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 3.

$$3x-5y=11,3x+3\times 3y=3\times 13$$

Whakarūnātia.

$$3x-5y=11,3x+9y=39$$

Me tango 3x+9y=39 mai i 3x-5y=11 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-5y-9y=11-39$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-5y-9y=11-39$$

Tāpiri -5y ki te -9y.

$$-14y=11-39$$

Tāpiri 11 ki te -39.

$$-14y=-28$$

Whakawehea ngā taha e rua ki te -14.

$$y=2$$

Whakaurua te 2 mō y ki x+3y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3\times 2=13$$

Whakareatia 3 ki te 2.

$$x+6=13$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=2$$