Whakaoti i te {l}{125x+110y=6100}{x+y=50}

$$\left\{ \begin{array} { l } { 125 x + 110 y = 6100 } \\ { x + y = 50 } \end{array} \right.$$

Whakaoti mō x, y

$$x=40$$

$$y=10$$

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.

$$125x+110y=6100,x+y=50$$

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

$$125x+110y=6100$$

Me tango 110y mai i ngā taha e rua o te whārite.

$$125x=-110y+6100$$

Whakawehea ngā taha e rua ki te 125.

$$x=\frac{1}{125}\left(-110y+6100\right)$$

Whakareatia \frac{1}{125} ki te -110y+6100.

$$x=-\frac{22}{25}y+\frac{244}{5}$$

Whakakapia te -\frac{22y}{25}+\frac{244}{5} mō te x ki tērā atu whārite, x+y=50.

$$-\frac{22}{25}y+\frac{244}{5}+y=50$$

Tāpiri -\frac{22y}{25} ki te y.

$$\frac{3}{25}y+\frac{244}{5}=50$$

Me tango \frac{244}{5} mai i ngā taha e rua o te whārite.

$$\frac{3}{25}y=\frac{6}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{25}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=10$$

Whakaurua te 10 mō y ki x=-\frac{22}{25}y+\frac{244}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{22}{25}\times 10+\frac{244}{5}$$

Whakareatia -\frac{22}{25} ki te 10.

$$x=\frac{-44+244}{5}$$

Tāpiri \frac{244}{5} ki te -\frac{44}{5} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=40$$

Kua oti te pūnaha te whakatau.

$$x=40,y=10$$

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.

$$125x+110y=6100,x+y=50$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}125&110\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6100\\50\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}125&110\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}125&110\\1&1\end{matrix}\right))\left(\begin{matrix}125&110\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}125&110\\1&1\end{matrix}\right))\left(\begin{matrix}6100\\50\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}125&110\\1&1\end{matrix}\right))\left(\begin{matrix}6100\\50\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}125&110\\1&1\end{matrix}\right))\left(\begin{matrix}6100\\50\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{125-110}&-\frac{110}{125-110}\\-\frac{1}{125-110}&\frac{125}{125-110}\end{matrix}\right)\left(\begin{matrix}6100\\50\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}&-\frac{22}{3}\\-\frac{1}{15}&\frac{25}{3}\end{matrix}\right)\left(\begin{matrix}6100\\50\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}\times 6100-\frac{22}{3}\times 50\\-\frac{1}{15}\times 6100+\frac{25}{3}\times 50\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=40,y=10$$

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.

$$125x+110y=6100,x+y=50$$

Kia ōrite ai a 125x 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 125.

$$125x+110y=6100,125x+125y=125\times 50$$

Whakarūnātia.

$$125x+110y=6100,125x+125y=6250$$

Me tango 125x+125y=6250 mai i 125x+110y=6100 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$125x-125x+110y-125y=6100-6250$$

Tāpiri 125x ki te -125x. Ka whakakore atu ngā kupu 125x me -125x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$110y-125y=6100-6250$$

Tāpiri 110y ki te -125y.

$$-15y=6100-6250$$

Tāpiri 6100 ki te -6250.

$$-15y=-150$$

Whakawehea ngā taha e rua ki te -15.

$$y=10$$

Whakaurua te 10 mō y ki x+y=50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+10=50$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$x=40$$

Kua oti te pūnaha te whakatau.

$$x=40,y=10$$