Whakaoti i te {c}{4x+3y=13}{x+3y=10}

$$\left. \begin{array} { c } { 4 x + 3 y = 13 } \\ { x + 3 y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=3$$

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.

$$4x+3y=13,x+3y=10$$

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

$$4x+3y=13$$

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

$$4x=-3y+13$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-3y+13\right)$$

Whakareatia \frac{1}{4} ki te -3y+13.

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

Whakakapia te \frac{-3y+13}{4} mō te x ki tērā atu whārite, x+3y=10.

$$-\frac{3}{4}y+\frac{13}{4}+3y=10$$

Tāpiri -\frac{3y}{4} ki te 3y.

$$\frac{9}{4}y+\frac{13}{4}=10$$

Me tango \frac{13}{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}{4}y+\frac{13}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\times 3+\frac{13}{4}$$

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

$$x=\frac{-9+13}{4}$$

Tāpiri \frac{13}{4} ki te -\frac{9}{4} 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.

$$x=1,y=3$$

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.

$$4x+3y=13,x+3y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&3\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&3\\1&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&3\\1&3\end{matrix}\right))\left(\begin{matrix}4&3\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\1&3\end{matrix}\right))\left(\begin{matrix}13\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\1&3\end{matrix}\right))\left(\begin{matrix}13\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\1&3\end{matrix}\right))\left(\begin{matrix}13\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4\times 3-3}&-\frac{3}{4\times 3-3}\\-\frac{1}{4\times 3-3}&\frac{4}{4\times 3-3}\end{matrix}\right)\left(\begin{matrix}13\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{1}{3}\\-\frac{1}{9}&\frac{4}{9}\end{matrix}\right)\left(\begin{matrix}13\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 13-\frac{1}{3}\times 10\\-\frac{1}{9}\times 13+\frac{4}{9}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=3$$

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.

$$4x+3y=13,x+3y=10$$

Me tango x+3y=10 mai i 4x+3y=13 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-x+3y-3y=13-10$$

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.

$$4x-x=13-10$$

Tāpiri 4x ki te -x.

$$3x=13-10$$

Tāpiri 13 ki te -10.

$$3x=3$$

Whakawehea ngā taha e rua ki te 3.

$$x=1$$

Whakaurua te 1 mō x ki x+3y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$1+3y=10$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$3y=9$$

Kua oti te pūnaha te whakatau.

$$x=1,y=3$$