Whakaoti i te {c}{4x+2y=50}{x+y=30}

$$\left. \begin{array} { c } { 4 x + 2 y = 50 } \\ { x + y = 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-5$$

$$y=35$$

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.

$$4x+2y=50,x+y=30$$

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

$$4x+2y=50$$

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

$$4x=-2y+50$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-2y+50\right)$$

Whakareatia \frac{1}{4} ki te -2y+50.

$$x=-\frac{1}{2}y+\frac{25}{2}$$

Whakakapia te \frac{-y+25}{2} mō te x ki tērā atu whārite, x+y=30.

$$-\frac{1}{2}y+\frac{25}{2}+y=30$$

Tāpiri -\frac{y}{2} ki te y.

$$\frac{1}{2}y+\frac{25}{2}=30$$

Me tango \frac{25}{2} mai i ngā taha e rua o te whārite.

$$\frac{1}{2}y=\frac{35}{2}$$

Me whakarea ngā taha e rua ki te 2.

$$y=35$$

Whakaurua te 35 mō y ki x=-\frac{1}{2}y+\frac{25}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 35+\frac{25}{2}$$

Whakareatia -\frac{1}{2} ki te 35.

$$x=\frac{-35+25}{2}$$

Tāpiri \frac{25}{2} ki te -\frac{35}{2} 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.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=35$$

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.

$$4x+2y=50,x+y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&2\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&2\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&2\\1&1\end{matrix}\right))\left(\begin{matrix}4&2\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\1&1\end{matrix}\right))\left(\begin{matrix}50\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\1&1\end{matrix}\right))\left(\begin{matrix}50\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\1&1\end{matrix}\right))\left(\begin{matrix}50\\30\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4-2}&-\frac{2}{4-2}\\-\frac{1}{4-2}&\frac{4}{4-2}\end{matrix}\right)\left(\begin{matrix}50\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-1\\-\frac{1}{2}&2\end{matrix}\right)\left(\begin{matrix}50\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 50-30\\-\frac{1}{2}\times 50+2\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\35\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-5,y=35$$

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.

$$4x+2y=50,x+y=30$$

Kia ōrite ai a 4x 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 4.

$$4x+2y=50,4x+4y=4\times 30$$

Whakarūnātia.

$$4x+2y=50,4x+4y=120$$

Me tango 4x+4y=120 mai i 4x+2y=50 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+2y-4y=50-120$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y-4y=50-120$$

Tāpiri 2y ki te -4y.

$$-2y=50-120$$

Tāpiri 50 ki te -120.

$$-2y=-70$$

Whakawehea ngā taha e rua ki te -2.

$$y=35$$

Whakaurua te 35 mō y ki x+y=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+35=30$$

Me tango 35 mai i ngā taha e rua o te whārite.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=35$$