Whakaoti i te {l}{10x+3y=3360}{x+y=420}

$$\left\{ \begin{array} { l } { 10 x + 3 y = 3360 } \\ { x + y = 420 } \end{array} \right.$$

Whakaoti mō x, y

$$x=300$$

$$y=120$$

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.

$$10x+3y=3360,x+y=420$$

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

$$10x+3y=3360$$

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

$$10x=-3y+3360$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(-3y+3360\right)$$

Whakareatia \frac{1}{10} ki te -3y+3360.

$$x=-\frac{3}{10}y+336$$

Whakakapia te -\frac{3y}{10}+336 mō te x ki tērā atu whārite, x+y=420.

$$-\frac{3}{10}y+336+y=420$$

Tāpiri -\frac{3y}{10} ki te y.

$$\frac{7}{10}y+336=420$$

Me tango 336 mai i ngā taha e rua o te whārite.

$$\frac{7}{10}y=84$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{10}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=120$$

Whakaurua te 120 mō y ki x=-\frac{3}{10}y+336. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{10}\times 120+336$$

Whakareatia -\frac{3}{10} ki te 120.

$$x=-36+336$$

Tāpiri 336 ki te -36.

$$x=300$$

Kua oti te pūnaha te whakatau.

$$x=300,y=120$$

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.

$$10x+3y=3360,x+y=420$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3360\\420\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&3\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&3\\1&1\end{matrix}\right))\left(\begin{matrix}10&3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&3\\1&1\end{matrix}\right))\left(\begin{matrix}3360\\420\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&3\\1&1\end{matrix}\right))\left(\begin{matrix}3360\\420\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&3\\1&1\end{matrix}\right))\left(\begin{matrix}3360\\420\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10-3}&-\frac{3}{10-3}\\-\frac{1}{10-3}&\frac{10}{10-3}\end{matrix}\right)\left(\begin{matrix}3360\\420\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&-\frac{3}{7}\\-\frac{1}{7}&\frac{10}{7}\end{matrix}\right)\left(\begin{matrix}3360\\420\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 3360-\frac{3}{7}\times 420\\-\frac{1}{7}\times 3360+\frac{10}{7}\times 420\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}300\\120\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=300,y=120$$

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.

$$10x+3y=3360,x+y=420$$

Kia ōrite ai a 10x 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 10.

$$10x+3y=3360,10x+10y=10\times 420$$

Whakarūnātia.

$$10x+3y=3360,10x+10y=4200$$

Me tango 10x+10y=4200 mai i 10x+3y=3360 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+3y-10y=3360-4200$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-10y=3360-4200$$

Tāpiri 3y ki te -10y.

$$-7y=3360-4200$$

Tāpiri 3360 ki te -4200.

$$-7y=-840$$

Whakawehea ngā taha e rua ki te -7.

$$y=120$$

Whakaurua te 120 mō y ki x+y=420. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+120=420$$

Me tango 120 mai i ngā taha e rua o te whārite.

$$x=300$$

Kua oti te pūnaha te whakatau.

$$x=300,y=120$$