Whakaoti i te {c}{3x+y=60}{x+y=40}

$$\left. \begin{array} { c } { 3 x + y = 60 } \\ { x + y = 40 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=30$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$3x+y=60,x+y=40$$

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

$$3x+y=60$$

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

$$3x=-y+60$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-y+60\right)$$

Whakareatia \frac{1}{3} ki te -y+60.

$$x=-\frac{1}{3}y+20$$

Whakakapia te -\frac{y}{3}+20 mō te x ki tērā atu whārite, x+y=40.

$$-\frac{1}{3}y+20+y=40$$

Tāpiri -\frac{y}{3} ki te y.

$$\frac{2}{3}y+20=40$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$\frac{2}{3}y=20$$

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=30$$

Whakaurua te 30 mō y ki x=-\frac{1}{3}y+20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times 30+20$$

Whakareatia -\frac{1}{3} ki te 30.

$$x=-10+20$$

Tāpiri 20 ki te -10.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=30$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$3x+y=60,x+y=40$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\40\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&1\\1&1\end{matrix}\right))\left(\begin{matrix}3&1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&1\end{matrix}\right))\left(\begin{matrix}60\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&1\end{matrix}\right))\left(\begin{matrix}60\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&1\end{matrix}\right))\left(\begin{matrix}60\\40\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-1}&-\frac{1}{3-1}\\-\frac{1}{3-1}&\frac{3}{3-1}\end{matrix}\right)\left(\begin{matrix}60\\40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{2}\\-\frac{1}{2}&\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}60\\40\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 60-\frac{1}{2}\times 40\\-\frac{1}{2}\times 60+\frac{3}{2}\times 40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\30\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=30$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$3x+y=60,x+y=40$$

Me tango x+y=40 mai i 3x+y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-x+y-y=60-40$$

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.

$$3x-x=60-40$$

Tāpiri 3x ki te -x.

$$2x=60-40$$

Tāpiri 60 ki te -40.

$$2x=20$$

Whakawehea ngā taha e rua ki te 2.

$$x=10$$

Whakaurua te 10 mō x ki x+y=40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$10+y=40$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$y=30$$

Kua oti te pūnaha te whakatau.

$$x=10,y=30$$