Whakaoti i te {l}{2x+3y=180}{x+2y=105}

$$\left. \begin{array} { l } { 2 x + 3 y = 180 } \\ { x + 2 y = 105 } \end{array} \right.$$

Whakaoti mō x, y

$$x=45$$

$$y=30$$

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.

$$2x+3y=180,x+2y=105$$

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

$$2x+3y=180$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+180$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+180\right)$$

Whakareatia \frac{1}{2} ki te -3y+180.

$$x=-\frac{3}{2}y+90$$

Whakakapia te -\frac{3y}{2}+90 mō te x ki tērā atu whārite, x+2y=105.

$$-\frac{3}{2}y+90+2y=105$$

Tāpiri -\frac{3y}{2} ki te 2y.

$$\frac{1}{2}y+90=105$$

Me tango 90 mai i ngā taha e rua o te whārite.

$$\frac{1}{2}y=15$$

Me whakarea ngā taha e rua ki te 2.

$$y=30$$

Whakaurua te 30 mō y ki x=-\frac{3}{2}y+90. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 30+90$$

Whakareatia -\frac{3}{2} ki te 30.

$$x=-45+90$$

Tāpiri 90 ki te -45.

$$x=45$$

Kua oti te pūnaha te whakatau.

$$x=45,y=30$$

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.

$$2x+3y=180,x+2y=105$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}180\\105\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}2&3\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}180\\105\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}180\\105\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}180\\105\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-3}&-\frac{3}{2\times 2-3}\\-\frac{1}{2\times 2-3}&\frac{2}{2\times 2-3}\end{matrix}\right)\left(\begin{matrix}180\\105\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-3\\-1&2\end{matrix}\right)\left(\begin{matrix}180\\105\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 180-3\times 105\\-180+2\times 105\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}45\\30\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=45,y=30$$

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.

$$2x+3y=180,x+2y=105$$

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.

$$2x+3y=180,2x+2\times 2y=2\times 105$$

Whakarūnātia.

$$2x+3y=180,2x+4y=210$$

Me tango 2x+4y=210 mai i 2x+3y=180 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+3y-4y=180-210$$

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.

$$3y-4y=180-210$$

Tāpiri 3y ki te -4y.

$$-y=180-210$$

Tāpiri 180 ki te -210.

$$-y=-30$$

Whakawehea ngā taha e rua ki te -1.

$$y=30$$

Whakaurua te 30 mō y ki x+2y=105. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+2\times 30=105$$

Whakareatia 2 ki te 30.

$$x+60=105$$

Me tango 60 mai i ngā taha e rua o te whārite.

$$x=45$$

Kua oti te pūnaha te whakatau.

$$x=45,y=30$$