Whakaoti i te {l}{2/3x+1/2y=5}{x-3y=6.}

$$\left\{ \begin{array} { l } { \frac { 2 } { 3 } x + \frac { 1 } { 2 } y = 5 } \\ { x - 3 y = 6 . } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{36}{5} = 7\frac{1}{5} = 7.2$$

$$y=\frac{2}{5}=0.4$$

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.

$$\frac{2}{3}x+\frac{1}{2}y=5,x-3y=6$$

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

$$\frac{2}{3}x+\frac{1}{2}y=5$$

Me tango \frac{y}{2} mai i ngā taha e rua o te whārite.

$$\frac{2}{3}x=-\frac{1}{2}y+5$$

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{3}{2}\left(-\frac{1}{2}y+5\right)$$

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

$$x=-\frac{3}{4}y+\frac{15}{2}$$

Whakakapia te -\frac{3y}{4}+\frac{15}{2} mō te x ki tērā atu whārite, x-3y=6.

$$-\frac{3}{4}y+\frac{15}{2}-3y=6$$

Tāpiri -\frac{3y}{4} ki te -3y.

$$-\frac{15}{4}y+\frac{15}{2}=6$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$-\frac{15}{4}y=-\frac{3}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{15}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{2}{5}$$

Whakaurua te \frac{2}{5} mō y ki x=-\frac{3}{4}y+\frac{15}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\times \frac{2}{5}+\frac{15}{2}$$

Whakareatia -\frac{3}{4} ki te \frac{2}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{3}{10}+\frac{15}{2}$$

Tāpiri \frac{15}{2} ki te -\frac{3}{10} 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.

$$x=\frac{36}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{36}{5},y=\frac{2}{5}$$

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.

$$\frac{2}{3}x+\frac{1}{2}y=5,x-3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{2}{3}&\frac{1}{2}\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{2}{3}&\frac{1}{2}\\1&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{2}{3}&\frac{1}{2}\\1&-3\end{matrix}\right))\left(\begin{matrix}\frac{2}{3}&\frac{1}{2}\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{2}{3}&\frac{1}{2}\\1&-3\end{matrix}\right))\left(\begin{matrix}5\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{2}{3}&\frac{1}{2}\\1&-3\end{matrix}\right))\left(\begin{matrix}5\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{2}{3}&\frac{1}{2}\\1&-3\end{matrix}\right))\left(\begin{matrix}5\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{\frac{2}{3}\left(-3\right)-\frac{1}{2}}&-\frac{\frac{1}{2}}{\frac{2}{3}\left(-3\right)-\frac{1}{2}}\\-\frac{1}{\frac{2}{3}\left(-3\right)-\frac{1}{2}}&\frac{\frac{2}{3}}{\frac{2}{3}\left(-3\right)-\frac{1}{2}}\end{matrix}\right)\left(\begin{matrix}5\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5}&\frac{1}{5}\\\frac{2}{5}&-\frac{4}{15}\end{matrix}\right)\left(\begin{matrix}5\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5}\times 5+\frac{1}{5}\times 6\\\frac{2}{5}\times 5-\frac{4}{15}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{36}{5}\\\frac{2}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{36}{5},y=\frac{2}{5}$$

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.

$$\frac{2}{3}x+\frac{1}{2}y=5,x-3y=6$$

Kia ōrite ai a \frac{2x}{3} 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 \frac{2}{3}.

$$\frac{2}{3}x+\frac{1}{2}y=5,\frac{2}{3}x+\frac{2}{3}\left(-3\right)y=\frac{2}{3}\times 6$$

Whakarūnātia.

$$\frac{2}{3}x+\frac{1}{2}y=5,\frac{2}{3}x-2y=4$$

Me tango \frac{2}{3}x-2y=4 mai i \frac{2}{3}x+\frac{1}{2}y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{2}{3}x-\frac{2}{3}x+\frac{1}{2}y+2y=5-4$$

Tāpiri \frac{2x}{3} ki te -\frac{2x}{3}. Ka whakakore atu ngā kupu \frac{2x}{3} me -\frac{2x}{3}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{1}{2}y+2y=5-4$$

Tāpiri \frac{y}{2} ki te 2y.

$$\frac{5}{2}y=5-4$$

Tāpiri 5 ki te -4.

$$\frac{5}{2}y=1$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{2}{5}$$

Whakaurua te \frac{2}{5} mō y ki x-3y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\times \frac{2}{5}=6$$

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

$$x-\frac{6}{5}=6$$

Me tāpiri \frac{6}{5} ki ngā taha e rua o te whārite.

$$x=\frac{36}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{36}{5},y=\frac{2}{5}$$