Whakaoti i te {l}{3x+7y=13}{5x-4y=6}

$$\left. \begin{array} { l } { 3 x + 7 y = 13 } \\ { 5 x - 4 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=1$$

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.

$$3x+7y=13,5x-4y=6$$

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

$$3x+7y=13$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$3x=-7y+13$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-7y+13\right)$$

Whakareatia \frac{1}{3} ki te -7y+13.

$$x=-\frac{7}{3}y+\frac{13}{3}$$

Whakakapia te \frac{-7y+13}{3} mō te x ki tērā atu whārite, 5x-4y=6.

$$5\left(-\frac{7}{3}y+\frac{13}{3}\right)-4y=6$$

Whakareatia 5 ki te \frac{-7y+13}{3}.

$$-\frac{35}{3}y+\frac{65}{3}-4y=6$$

Tāpiri -\frac{35y}{3} ki te -4y.

$$-\frac{47}{3}y+\frac{65}{3}=6$$

Me tango \frac{65}{3} mai i ngā taha e rua o te whārite.

$$-\frac{47}{3}y=-\frac{47}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{47}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=-\frac{7}{3}y+\frac{13}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{-7+13}{3}$$

Tāpiri \frac{13}{3} ki te -\frac{7}{3} 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.

$$x=2,y=1$$

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.

$$3x+7y=13,5x-4y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&7\\5&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\6\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&7\\5&-4\end{matrix}\right))\left(\begin{matrix}3&7\\5&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\5&-4\end{matrix}\right))\left(\begin{matrix}13\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\5&-4\end{matrix}\right))\left(\begin{matrix}13\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\5&-4\end{matrix}\right))\left(\begin{matrix}13\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{3\left(-4\right)-7\times 5}&-\frac{7}{3\left(-4\right)-7\times 5}\\-\frac{5}{3\left(-4\right)-7\times 5}&\frac{3}{3\left(-4\right)-7\times 5}\end{matrix}\right)\left(\begin{matrix}13\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{47}&\frac{7}{47}\\\frac{5}{47}&-\frac{3}{47}\end{matrix}\right)\left(\begin{matrix}13\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{47}\times 13+\frac{7}{47}\times 6\\\frac{5}{47}\times 13-\frac{3}{47}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=1$$

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.

$$3x+7y=13,5x-4y=6$$

Kia ōrite ai a 3x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$5\times 3x+5\times 7y=5\times 13,3\times 5x+3\left(-4\right)y=3\times 6$$

Whakarūnātia.

$$15x+35y=65,15x-12y=18$$

Me tango 15x-12y=18 mai i 15x+35y=65 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+35y+12y=65-18$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$35y+12y=65-18$$

Tāpiri 35y ki te 12y.

$$47y=65-18$$

Tāpiri 65 ki te -18.

$$47y=47$$

Whakawehea ngā taha e rua ki te 47.

$$y=1$$

Whakaurua te 1 mō y ki 5x-4y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-4=6$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$5x=10$$

Whakawehea ngā taha e rua ki te 5.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$