Whakaoti i te {l}{2x+3y-4=0}{x+3y=5}

$$\left\{ \begin{array} { l } { 2 x + 3 y - 4 = 0 } \\ { x + 3 y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=2$$

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.

$$2x+3y-4=0,x+3y=5$$

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

$$2x+3y-4=0$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$2x+3y=4$$

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

$$2x=-3y+4$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+4\right)$$

Whakareatia \frac{1}{2} ki te -3y+4.

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

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

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

Tāpiri -\frac{3y}{2} ki te 3y.

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

Me tango 2 mai i ngā taha e rua o te whārite.

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

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.

$$y=2$$

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

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

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

$$x=-3+2$$

Tāpiri 2 ki te -3.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=2$$

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.

$$2x+3y-4=0,x+3y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\1&3\end{matrix}\right))\left(\begin{matrix}2&3\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&3\end{matrix}\right))\left(\begin{matrix}4\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&3\end{matrix}\right))\left(\begin{matrix}4\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&3\end{matrix}\right))\left(\begin{matrix}4\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-3}&-\frac{3}{2\times 3-3}\\-\frac{1}{2\times 3-3}&\frac{2}{2\times 3-3}\end{matrix}\right)\left(\begin{matrix}4\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&-1\\-\frac{1}{3}&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}4\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4-5\\-\frac{1}{3}\times 4+\frac{2}{3}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=2$$

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.

$$2x+3y-4=0,x+3y=5$$

Me tango x+3y=5 mai i 2x+3y-4=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-x+3y-3y-4=-5$$

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.

$$2x-x-4=-5$$

Tāpiri 2x ki te -x.

$$x-4=-5$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=-1$$

Whakaurua te -1 mō x ki x+3y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-1+3y=5$$

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

$$3y=6$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=2$$