Whakaoti i te {l}{12x+3y=5}{3x+2y=70}

$$\left. \begin{array} { l } { 12 x + 3 y = 5 } \\ { 3 x + 2 y = 70 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{40}{3} = -13\frac{1}{3} \approx -13.333333333$$

$$y=55$$

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.

$$12x+3y=5,3x+2y=70$$

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

$$12x+3y=5$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$12x=-3y+5$$

Whakawehea ngā taha e rua ki te 12.

$$x=\frac{1}{12}\left(-3y+5\right)$$

Whakareatia \frac{1}{12} ki te -3y+5.

$$x=-\frac{1}{4}y+\frac{5}{12}$$

Whakakapia te -\frac{y}{4}+\frac{5}{12} mō te x ki tērā atu whārite, 3x+2y=70.

$$3\left(-\frac{1}{4}y+\frac{5}{12}\right)+2y=70$$

Whakareatia 3 ki te -\frac{y}{4}+\frac{5}{12}.

$$-\frac{3}{4}y+\frac{5}{4}+2y=70$$

Tāpiri -\frac{3y}{4} ki te 2y.

$$\frac{5}{4}y+\frac{5}{4}=70$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$\frac{5}{4}y=\frac{275}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=55$$

Whakaurua te 55 mō y ki x=-\frac{1}{4}y+\frac{5}{12}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{4}\times 55+\frac{5}{12}$$

Whakareatia -\frac{1}{4} ki te 55.

$$x=-\frac{55}{4}+\frac{5}{12}$$

Tāpiri \frac{5}{12} ki te -\frac{55}{4} 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.

$$x=-\frac{40}{3}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{40}{3},y=55$$

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.

$$12x+3y=5,3x+2y=70$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}12&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\70\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}12&3\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}12&3\\3&2\end{matrix}\right))\left(\begin{matrix}12&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{12\times 2-3\times 3}&-\frac{3}{12\times 2-3\times 3}\\-\frac{3}{12\times 2-3\times 3}&\frac{12}{12\times 2-3\times 3}\end{matrix}\right)\left(\begin{matrix}5\\70\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{15}&-\frac{1}{5}\\-\frac{1}{5}&\frac{4}{5}\end{matrix}\right)\left(\begin{matrix}5\\70\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{15}\times 5-\frac{1}{5}\times 70\\-\frac{1}{5}\times 5+\frac{4}{5}\times 70\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{40}{3}\\55\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{40}{3},y=55$$

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.

$$12x+3y=5,3x+2y=70$$

Kia ōrite ai a 12x 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 12.

$$3\times 12x+3\times 3y=3\times 5,12\times 3x+12\times 2y=12\times 70$$

Whakarūnātia.

$$36x+9y=15,36x+24y=840$$

Me tango 36x+24y=840 mai i 36x+9y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$36x-36x+9y-24y=15-840$$

Tāpiri 36x ki te -36x. Ka whakakore atu ngā kupu 36x me -36x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y-24y=15-840$$

Tāpiri 9y ki te -24y.

$$-15y=15-840$$

Tāpiri 15 ki te -840.

$$-15y=-825$$

Whakawehea ngā taha e rua ki te -15.

$$y=55$$

Whakaurua te 55 mō y ki 3x+2y=70. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times 55=70$$

Whakareatia 2 ki te 55.

$$3x+110=70$$

Me tango 110 mai i ngā taha e rua o te whārite.

$$3x=-40$$

Whakawehea ngā taha e rua ki te 3.

$$x=-\frac{40}{3}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{40}{3},y=55$$