Whakaoti i te {l}{73x-7y=66}{18x+98y=25}

$$\left. \begin{array} { l } { 73 x - 7 y = 66 } \\ { 18 x + 98 y = 25 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{73}{80}=0.9125$$

$$y=\frac{7}{80}=0.0875$$

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.

$$73x-7y=66,18x+98y=25$$

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

$$73x-7y=66$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$73x=7y+66$$

Whakawehea ngā taha e rua ki te 73.

$$x=\frac{1}{73}\left(7y+66\right)$$

Whakareatia \frac{1}{73} ki te 7y+66.

$$x=\frac{7}{73}y+\frac{66}{73}$$

Whakakapia te \frac{7y+66}{73} mō te x ki tērā atu whārite, 18x+98y=25.

$$18\left(\frac{7}{73}y+\frac{66}{73}\right)+98y=25$$

Whakareatia 18 ki te \frac{7y+66}{73}.

$$\frac{126}{73}y+\frac{1188}{73}+98y=25$$

Tāpiri \frac{126y}{73} ki te 98y.

$$\frac{7280}{73}y+\frac{1188}{73}=25$$

Me tango \frac{1188}{73} mai i ngā taha e rua o te whārite.

$$\frac{7280}{73}y=\frac{637}{73}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7280}{73}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{7}{80}$$

Whakaurua te \frac{7}{80} mō y ki x=\frac{7}{73}y+\frac{66}{73}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{73}\times \frac{7}{80}+\frac{66}{73}$$

Whakareatia \frac{7}{73} ki te \frac{7}{80} 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.

$$x=\frac{49}{5840}+\frac{66}{73}$$

Tāpiri \frac{66}{73} ki te \frac{49}{5840} 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.

$$x=\frac{73}{80}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{73}{80},y=\frac{7}{80}$$

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.

$$73x-7y=66,18x+98y=25$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}73&-7\\18&98\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}66\\25\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}73&-7\\18&98\end{matrix}\right).

$$inverse(\left(\begin{matrix}73&-7\\18&98\end{matrix}\right))\left(\begin{matrix}73&-7\\18&98\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}73&-7\\18&98\end{matrix}\right))\left(\begin{matrix}66\\25\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}73&-7\\18&98\end{matrix}\right))\left(\begin{matrix}66\\25\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}73&-7\\18&98\end{matrix}\right))\left(\begin{matrix}66\\25\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{98}{73\times 98-\left(-7\times 18\right)}&-\frac{-7}{73\times 98-\left(-7\times 18\right)}\\-\frac{18}{73\times 98-\left(-7\times 18\right)}&\frac{73}{73\times 98-\left(-7\times 18\right)}\end{matrix}\right)\left(\begin{matrix}66\\25\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{520}&\frac{1}{1040}\\-\frac{9}{3640}&\frac{73}{7280}\end{matrix}\right)\left(\begin{matrix}66\\25\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{520}\times 66+\frac{1}{1040}\times 25\\-\frac{9}{3640}\times 66+\frac{73}{7280}\times 25\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{73}{80}\\\frac{7}{80}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{73}{80},y=\frac{7}{80}$$

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.

$$73x-7y=66,18x+98y=25$$

Kia ōrite ai a 73x me 18x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 18 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 73.

$$18\times 73x+18\left(-7\right)y=18\times 66,73\times 18x+73\times 98y=73\times 25$$

Whakarūnātia.

$$1314x-126y=1188,1314x+7154y=1825$$

Me tango 1314x+7154y=1825 mai i 1314x-126y=1188 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$1314x-1314x-126y-7154y=1188-1825$$

Tāpiri 1314x ki te -1314x. Ka whakakore atu ngā kupu 1314x me -1314x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-126y-7154y=1188-1825$$

Tāpiri -126y ki te -7154y.

$$-7280y=1188-1825$$

Tāpiri 1188 ki te -1825.

$$-7280y=-637$$

Whakawehea ngā taha e rua ki te -7280.

$$y=\frac{7}{80}$$

Whakaurua te \frac{7}{80} mō y ki 18x+98y=25. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$18x+98\times \frac{7}{80}=25$$

Whakareatia 98 ki te \frac{7}{80}.

$$18x+\frac{343}{40}=25$$

Me tango \frac{343}{40} mai i ngā taha e rua o te whārite.

$$18x=\frac{657}{40}$$

Whakawehea ngā taha e rua ki te 18.

$$x=\frac{73}{80}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{73}{80},y=\frac{7}{80}$$