Whakaoti i te {l}{3x+y=14}{y=x-2}

$$\left\{ \begin{array} { l } { 3 x + y = 14 } \\ { y = x - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te x mai i ngā taha e rua.

$$y-x=-2$$

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.

$$3x+y=14,-x+y=-2$$

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

$$3x+y=14$$

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

$$3x=-y+14$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-y+14\right)$$

Whakareatia \frac{1}{3} ki te -y+14.

$$x=-\frac{1}{3}y+\frac{14}{3}$$

Whakakapia te \frac{-y+14}{3} mō te x ki tērā atu whārite, -x+y=-2.

$$-\left(-\frac{1}{3}y+\frac{14}{3}\right)+y=-2$$

Whakareatia -1 ki te \frac{-y+14}{3}.

$$\frac{1}{3}y-\frac{14}{3}+y=-2$$

Tāpiri \frac{y}{3} ki te y.

$$\frac{4}{3}y-\frac{14}{3}=-2$$

Me tāpiri \frac{14}{3} ki ngā taha e rua o te whārite.

$$\frac{4}{3}y=\frac{8}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{4}{3}, 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{1}{3}y+\frac{14}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times 2+\frac{14}{3}$$

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

$$x=\frac{-2+14}{3}$$

Tāpiri \frac{14}{3} ki te -\frac{2}{3} 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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te x mai i ngā taha e rua.

$$y-x=-2$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3x+y=14,-x+y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\-1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\-2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&1\\-1&1\end{matrix}\right))\left(\begin{matrix}3&1\\-1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\-1&1\end{matrix}\right))\left(\begin{matrix}14\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\-1&1\end{matrix}\right))\left(\begin{matrix}14\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\-1&1\end{matrix}\right))\left(\begin{matrix}14\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-\left(-1\right)}&-\frac{1}{3-\left(-1\right)}\\-\frac{-1}{3-\left(-1\right)}&\frac{3}{3-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}14\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&-\frac{1}{4}\\\frac{1}{4}&\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}14\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 14-\frac{1}{4}\left(-2\right)\\\frac{1}{4}\times 14+\frac{3}{4}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=4,y=2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te x mai i ngā taha e rua.

$$y-x=-2$$

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.

$$3x+y=14,-x+y=-2$$

Me tango -x+y=-2 mai i 3x+y=14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x+x+y-y=14+2$$

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

$$3x+x=14+2$$

Tāpiri 3x ki te x.

$$4x=14+2$$

Tāpiri 14 ki te 2.

$$4x=16$$

Whakawehea ngā taha e rua ki te 4.

$$x=4$$

Whakaurua te 4 mō x ki -x+y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-4+y=-2$$

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

$$y=2$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$