Whakaoti i te {l}{220x+108+100y=352}{600y+220x+108=316}

$$\left\{ \begin{array} { l } { 220 x + 108 + 100 y = 352 } \\ { 600 y + 220 x + 108 = 316 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{314}{275} = 1\frac{39}{275} \approx 1.141818182$$

$$y=-\frac{9}{125}=-0.072$$

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.

$$220x+100y+108=352,220x+600y+108=316$$

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

$$220x+100y+108=352$$

Me tango 108 mai i ngā taha e rua o te whārite.

$$220x+100y=244$$

Me tango 100y mai i ngā taha e rua o te whārite.

$$220x=-100y+244$$

Whakawehea ngā taha e rua ki te 220.

$$x=\frac{1}{220}\left(-100y+244\right)$$

Whakareatia \frac{1}{220} ki te -100y+244.

$$x=-\frac{5}{11}y+\frac{61}{55}$$

Whakakapia te -\frac{5y}{11}+\frac{61}{55} mō te x ki tērā atu whārite, 220x+600y+108=316.

$$220\left(-\frac{5}{11}y+\frac{61}{55}\right)+600y+108=316$$

Whakareatia 220 ki te -\frac{5y}{11}+\frac{61}{55}.

$$-100y+244+600y+108=316$$

Tāpiri -100y ki te 600y.

$$500y+244+108=316$$

Tāpiri 244 ki te 108.

$$500y+352=316$$

Me tango 352 mai i ngā taha e rua o te whārite.

$$500y=-36$$

Whakawehea ngā taha e rua ki te 500.

$$y=-\frac{9}{125}$$

Whakaurua te -\frac{9}{125} mō y ki x=-\frac{5}{11}y+\frac{61}{55}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{11}\left(-\frac{9}{125}\right)+\frac{61}{55}$$

Whakareatia -\frac{5}{11} ki te -\frac{9}{125} 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.

$$x=\frac{9}{275}+\frac{61}{55}$$

Tāpiri \frac{61}{55} ki te \frac{9}{275} 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.

$$x=\frac{314}{275}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{314}{275},y=-\frac{9}{125}$$

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.

$$220x+100y+108=352,220x+600y+108=316$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}220&100\\220&600\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}244\\208\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}220&100\\220&600\end{matrix}\right).

$$inverse(\left(\begin{matrix}220&100\\220&600\end{matrix}\right))\left(\begin{matrix}220&100\\220&600\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}220&100\\220&600\end{matrix}\right))\left(\begin{matrix}244\\208\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}220&100\\220&600\end{matrix}\right))\left(\begin{matrix}244\\208\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}220&100\\220&600\end{matrix}\right))\left(\begin{matrix}244\\208\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{600}{220\times 600-100\times 220}&-\frac{100}{220\times 600-100\times 220}\\-\frac{220}{220\times 600-100\times 220}&\frac{220}{220\times 600-100\times 220}\end{matrix}\right)\left(\begin{matrix}244\\208\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{550}&-\frac{1}{1100}\\-\frac{1}{500}&\frac{1}{500}\end{matrix}\right)\left(\begin{matrix}244\\208\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{550}\times 244-\frac{1}{1100}\times 208\\-\frac{1}{500}\times 244+\frac{1}{500}\times 208\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{314}{275}\\-\frac{9}{125}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{314}{275},y=-\frac{9}{125}$$

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.

$$220x+100y+108=352,220x+600y+108=316$$

Me tango 220x+600y+108=316 mai i 220x+100y+108=352 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$220x-220x+100y-600y+108-108=352-316$$

Tāpiri 220x ki te -220x. Ka whakakore atu ngā kupu 220x me -220x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$100y-600y+108-108=352-316$$

Tāpiri 100y ki te -600y.

$$-500y+108-108=352-316$$

Tāpiri 108 ki te -108.

$$-500y=352-316$$

Tāpiri 352 ki te -316.

$$-500y=36$$

Whakawehea ngā taha e rua ki te -500.

$$y=-\frac{9}{125}$$

Whakaurua te -\frac{9}{125} mō y ki 220x+600y+108=316. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$220x+600\left(-\frac{9}{125}\right)+108=316$$

Whakareatia 600 ki te -\frac{9}{125}.

$$220x-\frac{216}{5}+108=316$$

Tāpiri -\frac{216}{5} ki te 108.

$$220x+\frac{324}{5}=316$$

Me tango \frac{324}{5} mai i ngā taha e rua o te whārite.

$$220x=\frac{1256}{5}$$

Whakawehea ngā taha e rua ki te 220.

$$x=\frac{314}{275}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{314}{275},y=-\frac{9}{125}$$