Whakaoti i te {l}{6x+15y=360}{8x+10y=440}

$$\left\{ \begin{array} { l } { 6 x + 15 y = 360 } \\ { 8 x + 10 y = 440 } \end{array} \right.$$

Whakaoti mō x, y

$$x=50$$

$$y=4$$

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.

$$6x+15y=360,8x+10y=440$$

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

$$6x+15y=360$$

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

$$6x=-15y+360$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-15y+360\right)$$

Whakareatia \frac{1}{6} ki te -15y+360.

$$x=-\frac{5}{2}y+60$$

Whakakapia te -\frac{5y}{2}+60 mō te x ki tērā atu whārite, 8x+10y=440.

$$8\left(-\frac{5}{2}y+60\right)+10y=440$$

Whakareatia 8 ki te -\frac{5y}{2}+60.

$$-20y+480+10y=440$$

Tāpiri -20y ki te 10y.

$$-10y+480=440$$

Me tango 480 mai i ngā taha e rua o te whārite.

$$-10y=-40$$

Whakawehea ngā taha e rua ki te -10.

$$y=4$$

Whakaurua te 4 mō y ki x=-\frac{5}{2}y+60. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}\times 4+60$$

Whakareatia -\frac{5}{2} ki te 4.

$$x=-10+60$$

Tāpiri 60 ki te -10.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=4$$

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.

$$6x+15y=360,8x+10y=440$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&15\\8&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}360\\440\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&15\\8&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&15\\8&10\end{matrix}\right))\left(\begin{matrix}6&15\\8&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&15\\8&10\end{matrix}\right))\left(\begin{matrix}360\\440\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&15\\8&10\end{matrix}\right))\left(\begin{matrix}360\\440\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&15\\8&10\end{matrix}\right))\left(\begin{matrix}360\\440\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{6\times 10-15\times 8}&-\frac{15}{6\times 10-15\times 8}\\-\frac{8}{6\times 10-15\times 8}&\frac{6}{6\times 10-15\times 8}\end{matrix}\right)\left(\begin{matrix}360\\440\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}&\frac{1}{4}\\\frac{2}{15}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}360\\440\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}\times 360+\frac{1}{4}\times 440\\\frac{2}{15}\times 360-\frac{1}{10}\times 440\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=50,y=4$$

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.

$$6x+15y=360,8x+10y=440$$

Kia ōrite ai a 6x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$8\times 6x+8\times 15y=8\times 360,6\times 8x+6\times 10y=6\times 440$$

Whakarūnātia.

$$48x+120y=2880,48x+60y=2640$$

Me tango 48x+60y=2640 mai i 48x+120y=2880 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$48x-48x+120y-60y=2880-2640$$

Tāpiri 48x ki te -48x. Ka whakakore atu ngā kupu 48x me -48x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$120y-60y=2880-2640$$

Tāpiri 120y ki te -60y.

$$60y=2880-2640$$

Tāpiri 2880 ki te -2640.

$$60y=240$$

Whakawehea ngā taha e rua ki te 60.

$$y=4$$

Whakaurua te 4 mō y ki 8x+10y=440. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x+10\times 4=440$$

Whakareatia 10 ki te 4.

$$8x+40=440$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$8x=400$$

Whakawehea ngā taha e rua ki te 8.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=4$$