Whakaoti i te {l}{8x+3y=25}{2x+3y=13}

$$\left\{ \begin{array} { l } { 8 x + 3 y = 25 } \\ { 2 x + 3 y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=3$$

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.

$$8x+3y=25,2x+3y=13$$

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

$$8x+3y=25$$

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

$$8x=-3y+25$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-3y+25\right)$$

Whakareatia \frac{1}{8} ki te -3y+25.

$$x=-\frac{3}{8}y+\frac{25}{8}$$

Whakakapia te \frac{-3y+25}{8} mō te x ki tērā atu whārite, 2x+3y=13.

$$2\left(-\frac{3}{8}y+\frac{25}{8}\right)+3y=13$$

Whakareatia 2 ki te \frac{-3y+25}{8}.

$$-\frac{3}{4}y+\frac{25}{4}+3y=13$$

Tāpiri -\frac{3y}{4} ki te 3y.

$$\frac{9}{4}y+\frac{25}{4}=13$$

Me tango \frac{25}{4} mai i ngā taha e rua o te whārite.

$$\frac{9}{4}y=\frac{27}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=-\frac{3}{8}y+\frac{25}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{8}\times 3+\frac{25}{8}$$

Whakareatia -\frac{3}{8} ki te 3.

$$x=\frac{-9+25}{8}$$

Tāpiri \frac{25}{8} ki te -\frac{9}{8} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$

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.

$$8x+3y=25,2x+3y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&3\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&3\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&3\\2&3\end{matrix}\right))\left(\begin{matrix}8&3\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&3\\2&3\end{matrix}\right))\left(\begin{matrix}25\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&3\\2&3\end{matrix}\right))\left(\begin{matrix}25\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&3\\2&3\end{matrix}\right))\left(\begin{matrix}25\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8\times 3-3\times 2}&-\frac{3}{8\times 3-3\times 2}\\-\frac{2}{8\times 3-3\times 2}&\frac{8}{8\times 3-3\times 2}\end{matrix}\right)\left(\begin{matrix}25\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&-\frac{1}{6}\\-\frac{1}{9}&\frac{4}{9}\end{matrix}\right)\left(\begin{matrix}25\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 25-\frac{1}{6}\times 13\\-\frac{1}{9}\times 25+\frac{4}{9}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=3$$

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.

$$8x+3y=25,2x+3y=13$$

Me tango 2x+3y=13 mai i 8x+3y=25 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-2x+3y-3y=25-13$$

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.

$$8x-2x=25-13$$

Tāpiri 8x ki te -2x.

$$6x=25-13$$

Tāpiri 25 ki te -13.

$$6x=12$$

Whakawehea ngā taha e rua ki te 6.

$$x=2$$

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

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

Whakareatia 2 ki te 2.

$$4+3y=13$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$3y=9$$

Whakawehea ngā taha e rua ki te 3.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$