Whakaoti i te {l}{6x+2y=300}{3x+5y=600}

$$\left\{ \begin{array} { l } { 6 x + 2 y = 300 } \\ { 3 x + 5 y = 600 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{25}{2} = 12\frac{1}{2} = 12.5$$

$$y = \frac{225}{2} = 112\frac{1}{2} = 112.5$$

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.

$$6x+2y=300,3x+5y=600$$

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

$$6x+2y=300$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$6x=-2y+300$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-2y+300\right)$$

Whakareatia \frac{1}{6} ki te -2y+300.

$$x=-\frac{1}{3}y+50$$

Whakakapia te -\frac{y}{3}+50 mō te x ki tērā atu whārite, 3x+5y=600.

$$3\left(-\frac{1}{3}y+50\right)+5y=600$$

Whakareatia 3 ki te -\frac{y}{3}+50.

$$-y+150+5y=600$$

Tāpiri -y ki te 5y.

$$4y+150=600$$

Me tango 150 mai i ngā taha e rua o te whārite.

$$4y=450$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{225}{2}$$

Whakaurua te \frac{225}{2} mō y ki x=-\frac{1}{3}y+50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times \frac{225}{2}+50$$

Whakareatia -\frac{1}{3} ki te \frac{225}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{75}{2}+50$$

Tāpiri 50 ki te -\frac{75}{2}.

$$x=\frac{25}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{25}{2},y=\frac{225}{2}$$

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.

$$6x+2y=300,3x+5y=600$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&2\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}300\\600\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&2\\3&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&2\\3&5\end{matrix}\right))\left(\begin{matrix}6&2\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\3&5\end{matrix}\right))\left(\begin{matrix}300\\600\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\3&5\end{matrix}\right))\left(\begin{matrix}300\\600\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\3&5\end{matrix}\right))\left(\begin{matrix}300\\600\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6\times 5-2\times 3}&-\frac{2}{6\times 5-2\times 3}\\-\frac{3}{6\times 5-2\times 3}&\frac{6}{6\times 5-2\times 3}\end{matrix}\right)\left(\begin{matrix}300\\600\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{24}&-\frac{1}{12}\\-\frac{1}{8}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}300\\600\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{24}\times 300-\frac{1}{12}\times 600\\-\frac{1}{8}\times 300+\frac{1}{4}\times 600\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{2}\\\frac{225}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{25}{2},y=\frac{225}{2}$$

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.

$$6x+2y=300,3x+5y=600$$

Kia ōrite ai a 6x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$3\times 6x+3\times 2y=3\times 300,6\times 3x+6\times 5y=6\times 600$$

Whakarūnātia.

$$18x+6y=900,18x+30y=3600$$

Me tango 18x+30y=3600 mai i 18x+6y=900 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x+6y-30y=900-3600$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-30y=900-3600$$

Tāpiri 6y ki te -30y.

$$-24y=900-3600$$

Tāpiri 900 ki te -3600.

$$-24y=-2700$$

Whakawehea ngā taha e rua ki te -24.

$$y=\frac{225}{2}$$

Whakaurua te \frac{225}{2} mō y ki 3x+5y=600. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+5\times \frac{225}{2}=600$$

Whakareatia 5 ki te \frac{225}{2}.

$$3x+\frac{1125}{2}=600$$

Me tango \frac{1125}{2} mai i ngā taha e rua o te whārite.

$$3x=\frac{75}{2}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{25}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{25}{2},y=\frac{225}{2}$$