Whakaoti i te {l}{1/2x-2/3y-5=0}{3y+x=6}

$$\left. \begin{array} { l } { \frac { 1 } { 2 } x - \frac { 2 } { 3 } y - 5 = 0 } \\ { 3 y + x = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{114}{13} = 8\frac{10}{13} \approx 8.769230769$$

$$y=-\frac{12}{13}\approx -0.923076923$$

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.

$$\frac{1}{2}x-\frac{2}{3}y-5=0,x+3y=6$$

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

$$\frac{1}{2}x-\frac{2}{3}y-5=0$$

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

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

Me tāpiri \frac{2y}{3} ki ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te 2.

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

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

$$x=\frac{4}{3}y+10$$

Whakakapia te \frac{4y}{3}+10 mō te x ki tērā atu whārite, x+3y=6.

$$\frac{4}{3}y+10+3y=6$$

Tāpiri \frac{4y}{3} ki te 3y.

$$\frac{13}{3}y+10=6$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$\frac{13}{3}y=-4$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{12}{13}$$

Whakaurua te -\frac{12}{13} mō y ki x=\frac{4}{3}y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4}{3}\left(-\frac{12}{13}\right)+10$$

Whakareatia \frac{4}{3} ki te -\frac{12}{13} 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.

$$x=-\frac{16}{13}+10$$

Tāpiri 10 ki te -\frac{16}{13}.

$$x=\frac{114}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{114}{13},y=-\frac{12}{13}$$

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.

$$\frac{1}{2}x-\frac{2}{3}y-5=0,x+3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{13}&\frac{4}{13}\\-\frac{6}{13}&\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}5\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{13}\times 5+\frac{4}{13}\times 6\\-\frac{6}{13}\times 5+\frac{3}{13}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{114}{13}\\-\frac{12}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{114}{13},y=-\frac{12}{13}$$

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.

$$\frac{1}{2}x-\frac{2}{3}y-5=0,x+3y=6$$

Kia ōrite ai a \frac{x}{2} 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{1}{2}.

$$\frac{1}{2}x-\frac{2}{3}y-5=0,\frac{1}{2}x+\frac{1}{2}\times 3y=\frac{1}{2}\times 6$$

Whakarūnātia.

$$\frac{1}{2}x-\frac{2}{3}y-5=0,\frac{1}{2}x+\frac{3}{2}y=3$$

Me tango \frac{1}{2}x+\frac{3}{2}y=3 mai i \frac{1}{2}x-\frac{2}{3}y-5=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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

$$-\frac{2}{3}y-\frac{3}{2}y-5=-3$$

Tāpiri -\frac{2y}{3} ki te -\frac{3y}{2}.

$$-\frac{13}{6}y-5=-3$$

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

$$-\frac{13}{6}y=2$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{6}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{12}{13}$$

Whakaurua te -\frac{12}{13} 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.

$$x+3\left(-\frac{12}{13}\right)=6$$

Whakareatia 3 ki te -\frac{12}{13}.

$$x-\frac{36}{13}=6$$

Me tāpiri \frac{36}{13} ki ngā taha e rua o te whārite.

$$x=\frac{114}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{114}{13},y=-\frac{12}{13}$$