Whakaoti i te y=1/3x;quady=60-3x

$$y = \frac { 1 } { 3 } x ; \quad y = 60 - 3 x$$

Whakaoti mō y, x

$$x=18$$

$$y=6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-\frac{1}{3}x=0$$

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=60$$

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.

$$y-\frac{1}{3}x=0,y+3x=60$$

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

$$y-\frac{1}{3}x=0$$

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

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

Whakakapia te \frac{x}{3} mō te y ki tērā atu whārite, y+3x=60.

$$\frac{1}{3}x+3x=60$$

Tāpiri \frac{x}{3} ki te 3x.

$$\frac{10}{3}x=60$$

Whakawehea ngā taha e rua o te whārite ki te \frac{10}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=18$$

Whakaurua te 18 mō x ki y=\frac{1}{3}x. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{1}{3}\times 18$$

Whakareatia \frac{1}{3} ki te 18.

$$y=6$$

Kua oti te pūnaha te whakatau.

$$y=6,x=18$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-\frac{1}{3}x=0$$

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=60$$

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

$$y-\frac{1}{3}x=0,y+3x=60$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\60\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right))\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right))\left(\begin{matrix}0\\60\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right))\left(\begin{matrix}0\\60\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right))\left(\begin{matrix}0\\60\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-\left(-\frac{1}{3}\right)}&-\frac{-\frac{1}{3}}{3-\left(-\frac{1}{3}\right)}\\-\frac{1}{3-\left(-\frac{1}{3}\right)}&\frac{1}{3-\left(-\frac{1}{3}\right)}\end{matrix}\right)\left(\begin{matrix}0\\60\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{9}{10}&\frac{1}{10}\\-\frac{3}{10}&\frac{3}{10}\end{matrix}\right)\left(\begin{matrix}0\\60\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\times 60\\\frac{3}{10}\times 60\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\18\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=6,x=18$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-\frac{1}{3}x=0$$

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=60$$

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.

$$y-\frac{1}{3}x=0,y+3x=60$$

Me tango y+3x=60 mai i y-\frac{1}{3}x=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-\frac{1}{3}x-3x=-60$$

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.

$$-\frac{1}{3}x-3x=-60$$

Tāpiri -\frac{x}{3} ki te -3x.

$$-\frac{10}{3}x=-60$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{10}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=18$$

Whakaurua te 18 mō x ki y+3x=60. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+3\times 18=60$$

Whakareatia 3 ki te 18.

$$y+54=60$$

Me tango 54 mai i ngā taha e rua o te whārite.

$$y=6$$

Kua oti te pūnaha te whakatau.

$$y=6,x=18$$