Whakaoti i te {c}{2x+3y=5}{x+3y=6}

$$\left\{ \begin{array} { c } { 2 x + 3 y = 5 } \\ { x + 3 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

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.

$$2x+3y=5,x+3y=6$$

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

$$2x+3y=5$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+5$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -3y+5.

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

Whakakapia te \frac{-3y+5}{2} mō te x ki tērā atu whārite, x+3y=6.

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

Tāpiri -\frac{3y}{2} ki te 3y.

$$\frac{3}{2}y+\frac{5}{2}=6$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$\frac{3}{2}y=\frac{7}{2}$$

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

$$y=\frac{7}{3}$$

Whakaurua te \frac{7}{3} mō y ki x=-\frac{3}{2}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times \frac{7}{3}+\frac{5}{2}$$

Whakareatia -\frac{3}{2} ki te \frac{7}{3} 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.

$$x=\frac{-7+5}{2}$$

Tāpiri \frac{5}{2} ki te -\frac{7}{2} 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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=\frac{7}{3}$$

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.

$$2x+3y=5,x+3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\\frac{7}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=\frac{7}{3}$$

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.

$$2x+3y=5,x+3y=6$$

Me tango x+3y=6 mai i 2x+3y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-x+3y-3y=5-6$$

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

$$2x-x=5-6$$

Tāpiri 2x ki te -x.

$$x=5-6$$

Tāpiri 5 ki te -6.

$$x=-1$$

Whakaurua te -1 mō x ki x+3y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-1+3y=6$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$3y=7$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=\frac{7}{3}$$