Whakaoti i te {l}{x-y=10}{2x+(2y+1/2)=200}

$$\left\{ \begin{array} { l } { x - y = 10 } \\ { 2 x + ( 2 y + \frac { 1 } { 2 } ) = 200 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{439}{8} = 54\frac{7}{8} = 54.875$$

$$y = \frac{359}{8} = 44\frac{7}{8} = 44.875$$

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.

$$x-y=10,2x+2y+\frac{1}{2}=200$$

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

$$x-y=10$$

Me tāpiri y ki ngā taha e rua o te whārite.

$$x=y+10$$

Whakakapia te y+10 mō te x ki tērā atu whārite, 2x+2y+\frac{1}{2}=200.

$$2\left(y+10\right)+2y+\frac{1}{2}=200$$

Whakareatia 2 ki te y+10.

$$2y+20+2y+\frac{1}{2}=200$$

Tāpiri 2y ki te 2y.

$$4y+20+\frac{1}{2}=200$$

Tāpiri 20 ki te \frac{1}{2}.

$$4y+\frac{41}{2}=200$$

Me tango \frac{41}{2} mai i ngā taha e rua o te whārite.

$$4y=\frac{359}{2}$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{359}{8}$$

Whakaurua te \frac{359}{8} mō y ki x=y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{359}{8}+10$$

Tāpiri 10 ki te \frac{359}{8}.

$$x=\frac{439}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{439}{8},y=\frac{359}{8}$$

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.

$$x-y=10,2x+2y+\frac{1}{2}=200$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\\frac{399}{2}\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\2&2\end{matrix}\right))\left(\begin{matrix}1&-1\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&2\end{matrix}\right))\left(\begin{matrix}10\\\frac{399}{2}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&2\end{matrix}\right))\left(\begin{matrix}10\\\frac{399}{2}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&2\end{matrix}\right))\left(\begin{matrix}10\\\frac{399}{2}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-\left(-2\right)}&-\frac{-1}{2-\left(-2\right)}\\-\frac{2}{2-\left(-2\right)}&\frac{1}{2-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}10\\\frac{399}{2}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{4}\\-\frac{1}{2}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}10\\\frac{399}{2}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 10+\frac{1}{4}\times \frac{399}{2}\\-\frac{1}{2}\times 10+\frac{1}{4}\times \frac{399}{2}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{439}{8}\\\frac{359}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{439}{8},y=\frac{359}{8}$$

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.

$$x-y=10,2x+2y+\frac{1}{2}=200$$

Kia ōrite ai a x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2x+2\left(-1\right)y=2\times 10,2x+2y+\frac{1}{2}=200$$

Whakarūnātia.

$$2x-2y=20,2x+2y+\frac{1}{2}=200$$

Me tango 2x+2y+\frac{1}{2}=200 mai i 2x-2y=20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-2y-2y-\frac{1}{2}=20-200$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-2y-\frac{1}{2}=20-200$$

Tāpiri -2y ki te -2y.

$$-4y-\frac{1}{2}=20-200$$

Tāpiri 20 ki te -200.

$$-4y-\frac{1}{2}=-180$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$-4y=-\frac{359}{2}$$

Whakawehea ngā taha e rua ki te -4.

$$y=\frac{359}{8}$$

Whakaurua te \frac{359}{8} mō y ki 2x+2y+\frac{1}{2}=200. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+2\times \frac{359}{8}+\frac{1}{2}=200$$

Whakareatia 2 ki te \frac{359}{8}.

$$2x+\frac{359}{4}+\frac{1}{2}=200$$

Tāpiri \frac{359}{4} ki te \frac{1}{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.

$$2x+\frac{361}{4}=200$$

Me tango \frac{361}{4} mai i ngā taha e rua o te whārite.

$$2x=\frac{439}{4}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{439}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{439}{8},y=\frac{359}{8}$$