Whakaoti i te {c}{10x+14y=460}{x+y=40}

$$\left. \begin{array} { c } { 10 x + 14 y = 460 } \\ { x + y = 40 } \end{array} \right.$$

Whakaoti mō x, y

$$x=25$$

$$y=15$$

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.

$$10x+14y=460,x+y=40$$

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

$$10x+14y=460$$

Me tango 14y mai i ngā taha e rua o te whārite.

$$10x=-14y+460$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(-14y+460\right)$$

Whakareatia \frac{1}{10} ki te -14y+460.

$$x=-\frac{7}{5}y+46$$

Whakakapia te -\frac{7y}{5}+46 mō te x ki tērā atu whārite, x+y=40.

$$-\frac{7}{5}y+46+y=40$$

Tāpiri -\frac{7y}{5} ki te y.

$$-\frac{2}{5}y+46=40$$

Me tango 46 mai i ngā taha e rua o te whārite.

$$-\frac{2}{5}y=-6$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=15$$

Whakaurua te 15 mō y ki x=-\frac{7}{5}y+46. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{5}\times 15+46$$

Whakareatia -\frac{7}{5} ki te 15.

$$x=-21+46$$

Tāpiri 46 ki te -21.

$$x=25$$

Kua oti te pūnaha te whakatau.

$$x=25,y=15$$

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.

$$10x+14y=460,x+y=40$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&14\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}460\\40\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&14\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&14\\1&1\end{matrix}\right))\left(\begin{matrix}10&14\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&14\\1&1\end{matrix}\right))\left(\begin{matrix}460\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&14\\1&1\end{matrix}\right))\left(\begin{matrix}460\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&14\\1&1\end{matrix}\right))\left(\begin{matrix}460\\40\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10-14}&-\frac{14}{10-14}\\-\frac{1}{10-14}&\frac{10}{10-14}\end{matrix}\right)\left(\begin{matrix}460\\40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{7}{2}\\\frac{1}{4}&-\frac{5}{2}\end{matrix}\right)\left(\begin{matrix}460\\40\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 460+\frac{7}{2}\times 40\\\frac{1}{4}\times 460-\frac{5}{2}\times 40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\15\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=25,y=15$$

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.

$$10x+14y=460,x+y=40$$

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.

$$10x+14y=460,10x+10y=10\times 40$$

Whakarūnātia.

$$10x+14y=460,10x+10y=400$$

Me tango 10x+10y=400 mai i 10x+14y=460 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+14y-10y=460-400$$

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.

$$14y-10y=460-400$$

Tāpiri 14y ki te -10y.

$$4y=460-400$$

Tāpiri 460 ki te -400.

$$4y=60$$

Whakawehea ngā taha e rua ki te 4.

$$y=15$$

Whakaurua te 15 mō y ki x+y=40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+15=40$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$x=25$$

Kua oti te pūnaha te whakatau.

$$x=25,y=15$$