Whakaoti i te {l}{y=6x}{x+y+y=315.9}

$$\left. \begin{array} { l } { y = 6 x } \\ { x + y + y = 315.9 } \end{array} \right.$$

Whakaoti mō y, x

$$x=24.3$$

$$y=145.8$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=0$$

Whakaarohia te whārite tuarua. Pahekotia te y me y, ka 2y.

$$x+2y=315.9$$

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.

$$y-6x=0,2y+x=315.9$$

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

$$y-6x=0$$

Me tāpiri 6x ki ngā taha e rua o te whārite.

$$y=6x$$

Whakakapia te 6x mō te y ki tērā atu whārite, 2y+x=315.9.

$$2\times 6x+x=315.9$$

Whakareatia 2 ki te 6x.

$$12x+x=315.9$$

Tāpiri 12x ki te x.

$$13x=315.9$$

Whakawehea ngā taha e rua ki te 13.

$$x=24.3$$

Whakaurua te 24.3 mō x ki y=6x. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=6\times 24.3$$

Whakareatia 6 ki te 24.3.

$$y=145.8$$

Kua oti te pūnaha te whakatau.

$$y=145.8,x=24.3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=0$$

Whakaarohia te whārite tuarua. Pahekotia te y me y, ka 2y.

$$x+2y=315.9$$

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

$$y-6x=0,2y+x=315.9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-6\\2&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\315.9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-6\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-6\\2&1\end{matrix}\right))\left(\begin{matrix}1&-6\\2&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-6\\2&1\end{matrix}\right))\left(\begin{matrix}0\\315.9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-6\\2&1\end{matrix}\right))\left(\begin{matrix}0\\315.9\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-6\\2&1\end{matrix}\right))\left(\begin{matrix}0\\315.9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-6\times 2\right)}&-\frac{-6}{1-\left(-6\times 2\right)}\\-\frac{2}{1-\left(-6\times 2\right)}&\frac{1}{1-\left(-6\times 2\right)}\end{matrix}\right)\left(\begin{matrix}0\\315.9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}&\frac{6}{13}\\-\frac{2}{13}&\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}0\\315.9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{6}{13}\times 315.9\\\frac{1}{13}\times 315.9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{729}{5}\\\frac{243}{10}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{729}{5},x=\frac{243}{10}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=0$$

Whakaarohia te whārite tuarua. Pahekotia te y me y, ka 2y.

$$x+2y=315.9$$

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.

$$y-6x=0,2y+x=315.9$$

Kia ōrite ai a y me 2y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2y+2\left(-6\right)x=0,2y+x=315.9$$

Whakarūnātia.

$$2y-12x=0,2y+x=315.9$$

Me tango 2y+x=315.9 mai i 2y-12x=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2y-2y-12x-x=-315.9$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12x-x=-315.9$$

Tāpiri -12x ki te -x.

$$-13x=-315.9$$

Whakawehea ngā taha e rua ki te -13.

$$x=\frac{243}{10}$$

Whakaurua te \frac{243}{10} mō x ki 2y+x=315.9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y+\frac{243}{10}=315.9$$

Me tango \frac{243}{10} mai i ngā taha e rua o te whārite.

$$2y=\frac{1458}{5}$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{729}{5}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{729}{5},x=\frac{243}{10}$$