Whakaoti i te {l}{2x+y=60}{x=y}

$$\left\{ \begin{array} { l } { 2 x + y = 60 } \\ { x = y } \end{array} \right.$$

Whakaoti mō x, y

$$x=20$$

$$y=20$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-y=0$$

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.

$$2x+y=60,x-y=0$$

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

$$2x+y=60$$

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

$$2x=-y+60$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y+60\right)$$

Whakareatia \frac{1}{2} ki te -y+60.

$$x=-\frac{1}{2}y+30$$

Whakakapia te -\frac{y}{2}+30 mō te x ki tērā atu whārite, x-y=0.

$$-\frac{1}{2}y+30-y=0$$

Tāpiri -\frac{y}{2} ki te -y.

$$-\frac{3}{2}y+30=0$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$-\frac{3}{2}y=-30$$

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.

$$y=20$$

Whakaurua te 20 mō y ki x=-\frac{1}{2}y+30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 20+30$$

Whakareatia -\frac{1}{2} ki te 20.

$$x=-10+30$$

Tāpiri 30 ki te -10.

$$x=20$$

Kua oti te pūnaha te whakatau.

$$x=20,y=20$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-y=0$$

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

$$2x+y=60,x-y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\0\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&1\\1&-1\end{matrix}\right))\left(\begin{matrix}2&1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&-1\end{matrix}\right))\left(\begin{matrix}60\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&-1\end{matrix}\right))\left(\begin{matrix}60\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&-1\end{matrix}\right))\left(\begin{matrix}60\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-1}&-\frac{1}{2\left(-1\right)-1}\\-\frac{1}{2\left(-1\right)-1}&\frac{2}{2\left(-1\right)-1}\end{matrix}\right)\left(\begin{matrix}60\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{1}{3}\\\frac{1}{3}&-\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}60\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\20\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=20,y=20$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x-y=0$$

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.

$$2x+y=60,x-y=0$$

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

$$2x+y=60,2x+2\left(-1\right)y=0$$

Whakarūnātia.

$$2x+y=60,2x-2y=0$$

Me tango 2x-2y=0 mai i 2x+y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+y+2y=60$$

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

$$y+2y=60$$

Tāpiri y ki te 2y.

$$3y=60$$

Whakawehea ngā taha e rua ki te 3.

$$y=20$$

Whakaurua te 20 mō y ki x-y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-20=0$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$x=20$$

Kua oti te pūnaha te whakatau.

$$x=20,y=20$$