Whakaoti i te {l}{6x-5y=3}{3x+2y=12}

$$\left\{ \begin{array} { l } { 6 x - 5 y = 3 } \\ { 3 x + 2 y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{22}{9} = 2\frac{4}{9} \approx 2.444444444$$

$$y = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

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.

$$6x-5y=3,3x+2y=12$$

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

$$6x-5y=3$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$6x=5y+3$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(5y+3\right)$$

Whakareatia \frac{1}{6} ki te 5y+3.

$$x=\frac{5}{6}y+\frac{1}{2}$$

Whakakapia te \frac{5y}{6}+\frac{1}{2} mō te x ki tērā atu whārite, 3x+2y=12.

$$3\left(\frac{5}{6}y+\frac{1}{2}\right)+2y=12$$

Whakareatia 3 ki te \frac{5y}{6}+\frac{1}{2}.

$$\frac{5}{2}y+\frac{3}{2}+2y=12$$

Tāpiri \frac{5y}{2} ki te 2y.

$$\frac{9}{2}y+\frac{3}{2}=12$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$\frac{9}{2}y=\frac{21}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{7}{3}$$

Whakaurua te \frac{7}{3} mō y ki x=\frac{5}{6}y+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{6}\times \frac{7}{3}+\frac{1}{2}$$

Whakareatia \frac{5}{6} ki te \frac{7}{3} 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.

$$x=\frac{35}{18}+\frac{1}{2}$$

Tāpiri \frac{1}{2} ki te \frac{35}{18} 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.

$$x=\frac{22}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{22}{9},y=\frac{7}{3}$$

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.

$$6x-5y=3,3x+2y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&-5\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&-5\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&-5\\3&2\end{matrix}\right))\left(\begin{matrix}6&-5\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\3&2\end{matrix}\right))\left(\begin{matrix}3\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\3&2\end{matrix}\right))\left(\begin{matrix}3\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\3&2\end{matrix}\right))\left(\begin{matrix}3\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{6\times 2-\left(-5\times 3\right)}&-\frac{-5}{6\times 2-\left(-5\times 3\right)}\\-\frac{3}{6\times 2-\left(-5\times 3\right)}&\frac{6}{6\times 2-\left(-5\times 3\right)}\end{matrix}\right)\left(\begin{matrix}3\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{27}&\frac{5}{27}\\-\frac{1}{9}&\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}3\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{27}\times 3+\frac{5}{27}\times 12\\-\frac{1}{9}\times 3+\frac{2}{9}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{22}{9}\\\frac{7}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{22}{9},y=\frac{7}{3}$$

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.

$$6x-5y=3,3x+2y=12$$

Kia ōrite ai a 6x 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 6.

$$3\times 6x+3\left(-5\right)y=3\times 3,6\times 3x+6\times 2y=6\times 12$$

Whakarūnātia.

$$18x-15y=9,18x+12y=72$$

Me tango 18x+12y=72 mai i 18x-15y=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x-15y-12y=9-72$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-15y-12y=9-72$$

Tāpiri -15y ki te -12y.

$$-27y=9-72$$

Tāpiri 9 ki te -72.

$$-27y=-63$$

Whakawehea ngā taha e rua ki te -27.

$$y=\frac{7}{3}$$

Whakaurua te \frac{7}{3} mō y ki 3x+2y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times \frac{7}{3}=12$$

Whakareatia 2 ki te \frac{7}{3}.

$$3x+\frac{14}{3}=12$$

Me tango \frac{14}{3} mai i ngā taha e rua o te whārite.

$$3x=\frac{22}{3}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{22}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{22}{9},y=\frac{7}{3}$$