Whakaoti i te {l}{60x+60y=1170-300}{70x+140y=2035}

$$\left. \begin{array}{l}{ 60 x + 60 y = 1170 - 300 }\\{ 70 x + 140 y = 2035 }\end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{1}{14}\approx -0.071428571$$

$$y = \frac{102}{7} = 14\frac{4}{7} \approx 14.571428571$$

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.

$$60x+60y=870,70x+140y=2035$$

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

$$60x+60y=870$$

Me tango 60y mai i ngā taha e rua o te whārite.

$$60x=-60y+870$$

Whakawehea ngā taha e rua ki te 60.

$$x=\frac{1}{60}\left(-60y+870\right)$$

Whakareatia \frac{1}{60} ki te -60y+870.

$$x=-y+\frac{29}{2}$$

Whakakapia te -y+\frac{29}{2} mō te x ki tērā atu whārite, 70x+140y=2035.

$$70\left(-y+\frac{29}{2}\right)+140y=2035$$

Whakareatia 70 ki te -y+\frac{29}{2}.

$$-70y+1015+140y=2035$$

Tāpiri -70y ki te 140y.

$$70y+1015=2035$$

Me tango 1015 mai i ngā taha e rua o te whārite.

$$70y=1020$$

Whakawehea ngā taha e rua ki te 70.

$$y=\frac{102}{7}$$

Whakaurua te \frac{102}{7} mō y ki x=-y+\frac{29}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{102}{7}+\frac{29}{2}$$

Tāpiri \frac{29}{2} ki te -\frac{102}{7} 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.

$$x=-\frac{1}{14}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{14},y=\frac{102}{7}$$

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.

$$60x+60y=870,70x+140y=2035$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}60&60\\70&140\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}870\\2035\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}60&60\\70&140\end{matrix}\right).

$$inverse(\left(\begin{matrix}60&60\\70&140\end{matrix}\right))\left(\begin{matrix}60&60\\70&140\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&60\\70&140\end{matrix}\right))\left(\begin{matrix}870\\2035\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&60\\70&140\end{matrix}\right))\left(\begin{matrix}870\\2035\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&60\\70&140\end{matrix}\right))\left(\begin{matrix}870\\2035\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{140}{60\times 140-60\times 70}&-\frac{60}{60\times 140-60\times 70}\\-\frac{70}{60\times 140-60\times 70}&\frac{60}{60\times 140-60\times 70}\end{matrix}\right)\left(\begin{matrix}870\\2035\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{30}&-\frac{1}{70}\\-\frac{1}{60}&\frac{1}{70}\end{matrix}\right)\left(\begin{matrix}870\\2035\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{30}\times 870-\frac{1}{70}\times 2035\\-\frac{1}{60}\times 870+\frac{1}{70}\times 2035\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{14}\\\frac{102}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1}{14},y=\frac{102}{7}$$

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.

$$60x+60y=870,70x+140y=2035$$

Kia ōrite ai a 60x me 70x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 70 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 60.

$$70\times 60x+70\times 60y=70\times 870,60\times 70x+60\times 140y=60\times 2035$$

Whakarūnātia.

$$4200x+4200y=60900,4200x+8400y=122100$$

Me tango 4200x+8400y=122100 mai i 4200x+4200y=60900 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4200x-4200x+4200y-8400y=60900-122100$$

Tāpiri 4200x ki te -4200x. Ka whakakore atu ngā kupu 4200x me -4200x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4200y-8400y=60900-122100$$

Tāpiri 4200y ki te -8400y.

$$-4200y=60900-122100$$

Tāpiri 60900 ki te -122100.

$$-4200y=-61200$$

Whakawehea ngā taha e rua ki te -4200.

$$y=\frac{102}{7}$$

Whakaurua te \frac{102}{7} mō y ki 70x+140y=2035. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$70x+140\times \frac{102}{7}=2035$$

Whakareatia 140 ki te \frac{102}{7}.

$$70x+2040=2035$$

Me tango 2040 mai i ngā taha e rua o te whārite.

$$70x=-5$$

Whakawehea ngā taha e rua ki te 70.

$$x=-\frac{1}{14}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{14},y=\frac{102}{7}$$