Whakaoti i te {l}{2x+3y=100}{x+y=42}

$$\left. \begin{array} { l } { 2 x + 3 y = 100 } \\ { x + y = 42 } \end{array} \right.$$

Whakaoti mō x, y

$$x=26$$

$$y=16$$

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.

$$2x+3y=100,x+y=42$$

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

$$2x+3y=100$$

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

$$2x=-3y+100$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+100\right)$$

Whakareatia \frac{1}{2} ki te -3y+100.

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

Whakakapia te -\frac{3y}{2}+50 mō te x ki tērā atu whārite, x+y=42.

$$-\frac{3}{2}y+50+y=42$$

Tāpiri -\frac{3y}{2} ki te y.

$$-\frac{1}{2}y+50=42$$

Me tango 50 mai i ngā taha e rua o te whārite.

$$-\frac{1}{2}y=-8$$

Me whakarea ngā taha e rua ki te -2.

$$y=16$$

Whakaurua te 16 mō y ki x=-\frac{3}{2}y+50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 16+50$$

Whakareatia -\frac{3}{2} ki te 16.

$$x=-24+50$$

Tāpiri 50 ki te -24.

$$x=26$$

Kua oti te pūnaha te whakatau.

$$x=26,y=16$$

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.

$$2x+3y=100,x+y=42$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}100\\42\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\1&1\end{matrix}\right))\left(\begin{matrix}2&3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&1\end{matrix}\right))\left(\begin{matrix}100\\42\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&1\end{matrix}\right))\left(\begin{matrix}100\\42\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&1\end{matrix}\right))\left(\begin{matrix}100\\42\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-3}&-\frac{3}{2-3}\\-\frac{1}{2-3}&\frac{2}{2-3}\end{matrix}\right)\left(\begin{matrix}100\\42\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&3\\1&-2\end{matrix}\right)\left(\begin{matrix}100\\42\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-100+3\times 42\\100-2\times 42\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}26\\16\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=26,y=16$$

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.

$$2x+3y=100,x+y=42$$

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.

$$2x+3y=100,2x+2y=2\times 42$$

Whakarūnātia.

$$2x+3y=100,2x+2y=84$$

Me tango 2x+2y=84 mai i 2x+3y=100 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+3y-2y=100-84$$

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.

$$3y-2y=100-84$$

Tāpiri 3y ki te -2y.

$$y=100-84$$

Tāpiri 100 ki te -84.

$$y=16$$

Whakaurua te 16 mō y ki x+y=42. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+16=42$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$x=26$$

Kua oti te pūnaha te whakatau.

$$x=26,y=16$$