Whakaoti i te {l}{11x+5y=7}{6x+3y=21}

$$\left. \begin{array} { l } { 11 x + 5 y = 7 } \\ { 6 x + 3 y = 21 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-28$$

$$y=63$$

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.

$$11x+5y=7,6x+3y=21$$

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

$$11x+5y=7$$

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

$$11x=-5y+7$$

Whakawehea ngā taha e rua ki te 11.

$$x=\frac{1}{11}\left(-5y+7\right)$$

Whakareatia \frac{1}{11} ki te -5y+7.

$$x=-\frac{5}{11}y+\frac{7}{11}$$

Whakakapia te \frac{-5y+7}{11} mō te x ki tērā atu whārite, 6x+3y=21.

$$6\left(-\frac{5}{11}y+\frac{7}{11}\right)+3y=21$$

Whakareatia 6 ki te \frac{-5y+7}{11}.

$$-\frac{30}{11}y+\frac{42}{11}+3y=21$$

Tāpiri -\frac{30y}{11} ki te 3y.

$$\frac{3}{11}y+\frac{42}{11}=21$$

Me tango \frac{42}{11} mai i ngā taha e rua o te whārite.

$$\frac{3}{11}y=\frac{189}{11}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{11}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=63$$

Whakaurua te 63 mō y ki x=-\frac{5}{11}y+\frac{7}{11}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{11}\times 63+\frac{7}{11}$$

Whakareatia -\frac{5}{11} ki te 63.

$$x=\frac{-315+7}{11}$$

Tāpiri \frac{7}{11} ki te -\frac{315}{11} 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.

$$x=-28$$

Kua oti te pūnaha te whakatau.

$$x=-28,y=63$$

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.

$$11x+5y=7,6x+3y=21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}11&5\\6&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\21\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}11&5\\6&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}11&5\\6&3\end{matrix}\right))\left(\begin{matrix}11&5\\6&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&5\\6&3\end{matrix}\right))\left(\begin{matrix}7\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&5\\6&3\end{matrix}\right))\left(\begin{matrix}7\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&5\\6&3\end{matrix}\right))\left(\begin{matrix}7\\21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11\times 3-5\times 6}&-\frac{5}{11\times 3-5\times 6}\\-\frac{6}{11\times 3-5\times 6}&\frac{11}{11\times 3-5\times 6}\end{matrix}\right)\left(\begin{matrix}7\\21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&-\frac{5}{3}\\-2&\frac{11}{3}\end{matrix}\right)\left(\begin{matrix}7\\21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7-\frac{5}{3}\times 21\\-2\times 7+\frac{11}{3}\times 21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-28\\63\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-28,y=63$$

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.

$$11x+5y=7,6x+3y=21$$

Kia ōrite ai a 11x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 11.

$$6\times 11x+6\times 5y=6\times 7,11\times 6x+11\times 3y=11\times 21$$

Whakarūnātia.

$$66x+30y=42,66x+33y=231$$

Me tango 66x+33y=231 mai i 66x+30y=42 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$66x-66x+30y-33y=42-231$$

Tāpiri 66x ki te -66x. Ka whakakore atu ngā kupu 66x me -66x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$30y-33y=42-231$$

Tāpiri 30y ki te -33y.

$$-3y=42-231$$

Tāpiri 42 ki te -231.

$$-3y=-189$$

Whakawehea ngā taha e rua ki te -3.

$$y=63$$

Whakaurua te 63 mō y ki 6x+3y=21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+3\times 63=21$$

Whakareatia 3 ki te 63.

$$6x+189=21$$

Me tango 189 mai i ngā taha e rua o te whārite.

$$6x=-168$$

Whakawehea ngā taha e rua ki te 6.

$$x=-28$$

Kua oti te pūnaha te whakatau.

$$x=-28,y=63$$