Whakaoti i te {c}{7x+5y=54}{3x+4y=38}

$$\left. \begin{array} { c } { 7 x + 5 y = 54 } \\ { 3 x + 4 y = 38 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=8$$

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.

$$7x+5y=54,3x+4y=38$$

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

$$7x+5y=54$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$7x=-5y+54$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-5y+54\right)$$

Whakareatia \frac{1}{7} ki te -5y+54.

$$x=-\frac{5}{7}y+\frac{54}{7}$$

Whakakapia te \frac{-5y+54}{7} mō te x ki tērā atu whārite, 3x+4y=38.

$$3\left(-\frac{5}{7}y+\frac{54}{7}\right)+4y=38$$

Whakareatia 3 ki te \frac{-5y+54}{7}.

$$-\frac{15}{7}y+\frac{162}{7}+4y=38$$

Tāpiri -\frac{15y}{7} ki te 4y.

$$\frac{13}{7}y+\frac{162}{7}=38$$

Me tango \frac{162}{7} mai i ngā taha e rua o te whārite.

$$\frac{13}{7}y=\frac{104}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=8$$

Whakaurua te 8 mō y ki x=-\frac{5}{7}y+\frac{54}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{7}\times 8+\frac{54}{7}$$

Whakareatia -\frac{5}{7} ki te 8.

$$x=\frac{-40+54}{7}$$

Tāpiri \frac{54}{7} ki te -\frac{40}{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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=8$$

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.

$$7x+5y=54,3x+4y=38$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&5\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}54\\38\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&5\\3&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&5\\3&4\end{matrix}\right))\left(\begin{matrix}7&5\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\3&4\end{matrix}\right))\left(\begin{matrix}54\\38\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\3&4\end{matrix}\right))\left(\begin{matrix}54\\38\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\3&4\end{matrix}\right))\left(\begin{matrix}54\\38\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7\times 4-5\times 3}&-\frac{5}{7\times 4-5\times 3}\\-\frac{3}{7\times 4-5\times 3}&\frac{7}{7\times 4-5\times 3}\end{matrix}\right)\left(\begin{matrix}54\\38\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{13}&-\frac{5}{13}\\-\frac{3}{13}&\frac{7}{13}\end{matrix}\right)\left(\begin{matrix}54\\38\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{13}\times 54-\frac{5}{13}\times 38\\-\frac{3}{13}\times 54+\frac{7}{13}\times 38\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=8$$

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.

$$7x+5y=54,3x+4y=38$$

Kia ōrite ai a 7x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$3\times 7x+3\times 5y=3\times 54,7\times 3x+7\times 4y=7\times 38$$

Whakarūnātia.

$$21x+15y=162,21x+28y=266$$

Me tango 21x+28y=266 mai i 21x+15y=162 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$21x-21x+15y-28y=162-266$$

Tāpiri 21x ki te -21x. Ka whakakore atu ngā kupu 21x me -21x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y-28y=162-266$$

Tāpiri 15y ki te -28y.

$$-13y=162-266$$

Tāpiri 162 ki te -266.

$$-13y=-104$$

Whakawehea ngā taha e rua ki te -13.

$$y=8$$

Whakaurua te 8 mō y ki 3x+4y=38. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+4\times 8=38$$

Whakareatia 4 ki te 8.

$$3x+32=38$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=8$$