Whakaoti i te {c}{y=1/3x+7}{x+y=3}

$$\left. \begin{array} { c } { y = \frac { 1 } { 3 } x + 7 } \\ { x + y = 3 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-3$$

$$y=6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te \frac{1}{3}x mai i ngā taha e rua.

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

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.

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

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

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

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

$$y=\frac{1}{3}x+7$$

Whakakapia te \frac{x}{3}+7 mō te y ki tērā atu whārite, y+x=3.

$$\frac{1}{3}x+7+x=3$$

Tāpiri \frac{x}{3} ki te x.

$$\frac{4}{3}x+7=3$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$\frac{4}{3}x=-4$$

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.

$$x=-3$$

Whakaurua te -3 mō x ki y=\frac{1}{3}x+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{1}{3}\left(-3\right)+7$$

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

$$y=-1+7$$

Tāpiri 7 ki te -1.

$$y=6$$

Kua oti te pūnaha te whakatau.

$$y=6,x=-3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te \frac{1}{3}x mai i ngā taha e rua.

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

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&1\end{matrix}\right))\left(\begin{matrix}7\\3\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&\frac{1}{4}\\-\frac{3}{4}&\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}7\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\times 7+\frac{1}{4}\times 3\\-\frac{3}{4}\times 7+\frac{3}{4}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=6,x=-3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te \frac{1}{3}x mai i ngā taha e rua.

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

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.

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

Me tango y+x=3 mai i y-\frac{1}{3}x=7 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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.

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

Tāpiri -\frac{x}{3} ki te -x.

$$-\frac{4}{3}x=7-3$$

Tāpiri 7 ki te -3.

$$-\frac{4}{3}x=4$$

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.

$$x=-3$$

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

$$y-3=3$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$y=6$$

Kua oti te pūnaha te whakatau.

$$y=6,x=-3$$