Whakaoti i te {l}{3y+2x=75}{x+y=50}

$$\left. \begin{array} { l } { 3 y + 2 x = 75 } \\ { x + y = 50 } \end{array} \right.$$

Whakaoti mō y, x

$$x=75$$

$$y=-25$$

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.

$$3y+2x=75,y+x=50$$

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

$$3y+2x=75$$

Me tango 2x mai i ngā taha e rua o te whārite.

$$3y=-2x+75$$

Whakawehea ngā taha e rua ki te 3.

$$y=\frac{1}{3}\left(-2x+75\right)$$

Whakareatia \frac{1}{3} ki te -2x+75.

$$y=-\frac{2}{3}x+25$$

Whakakapia te -\frac{2x}{3}+25 mō te y ki tērā atu whārite, y+x=50.

$$-\frac{2}{3}x+25+x=50$$

Tāpiri -\frac{2x}{3} ki te x.

$$\frac{1}{3}x+25=50$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$\frac{1}{3}x=25$$

Me whakarea ngā taha e rua ki te 3.

$$x=75$$

Whakaurua te 75 mō x ki y=-\frac{2}{3}x+25. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{2}{3}\times 75+25$$

Whakareatia -\frac{2}{3} ki te 75.

$$y=-50+25$$

Tāpiri 25 ki te -50.

$$y=-25$$

Kua oti te pūnaha te whakatau.

$$y=-25,x=75$$

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.

$$3y+2x=75,y+x=50$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}75\\50\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&2\\1&1\end{matrix}\right))\left(\begin{matrix}3&2\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\1&1\end{matrix}\right))\left(\begin{matrix}75\\50\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\1&1\end{matrix}\right))\left(\begin{matrix}75\\50\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\1&1\end{matrix}\right))\left(\begin{matrix}75\\50\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-2}&-\frac{2}{3-2}\\-\frac{1}{3-2}&\frac{3}{3-2}\end{matrix}\right)\left(\begin{matrix}75\\50\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1&-2\\-1&3\end{matrix}\right)\left(\begin{matrix}75\\50\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}75-2\times 50\\-75+3\times 50\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-25\\75\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-25,x=75$$

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.

$$3y+2x=75,y+x=50$$

Kia ōrite ai a 3y me y, 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 3.

$$3y+2x=75,3y+3x=3\times 50$$

Whakarūnātia.

$$3y+2x=75,3y+3x=150$$

Me tango 3y+3x=150 mai i 3y+2x=75 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3y-3y+2x-3x=75-150$$

Tāpiri 3y ki te -3y. Ka whakakore atu ngā kupu 3y me -3y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2x-3x=75-150$$

Tāpiri 2x ki te -3x.

$$-x=75-150$$

Tāpiri 75 ki te -150.

$$-x=-75$$

Whakawehea ngā taha e rua ki te -1.

$$x=75$$

Whakaurua te 75 mō x ki y+x=50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+75=50$$

Me tango 75 mai i ngā taha e rua o te whārite.

$$y=-25$$

Kua oti te pūnaha te whakatau.

$$y=-25,x=75$$