Whakaoti i te {l}{6x-5y=-36}{-7x+2y=39}

$$\left. \begin{array} { l } { 6 x - 5 y = - 36 } \\ { - 7 x + 2 y = 39 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{123}{23} = -5\frac{8}{23} \approx -5.347826087$$

$$y=\frac{18}{23}\approx 0.782608696$$

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.

$$6x-5y=-36,-7x+2y=39$$

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

$$6x-5y=-36$$

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

$$6x=5y-36$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(5y-36\right)$$

Whakareatia \frac{1}{6} ki te 5y-36.

$$x=\frac{5}{6}y-6$$

Whakakapia te \frac{5y}{6}-6 mō te x ki tērā atu whārite, -7x+2y=39.

$$-7\left(\frac{5}{6}y-6\right)+2y=39$$

Whakareatia -7 ki te \frac{5y}{6}-6.

$$-\frac{35}{6}y+42+2y=39$$

Tāpiri -\frac{35y}{6} ki te 2y.

$$-\frac{23}{6}y+42=39$$

Me tango 42 mai i ngā taha e rua o te whārite.

$$-\frac{23}{6}y=-3$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{23}{6}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{18}{23}$$

Whakaurua te \frac{18}{23} mō y ki x=\frac{5}{6}y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{6}\times \frac{18}{23}-6$$

Whakareatia \frac{5}{6} ki te \frac{18}{23} 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.

$$x=\frac{15}{23}-6$$

Tāpiri -6 ki te \frac{15}{23}.

$$x=-\frac{123}{23}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{123}{23},y=\frac{18}{23}$$

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.

$$6x-5y=-36,-7x+2y=39$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&-5\\-7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-36\\39\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&-5\\-7&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&-5\\-7&2\end{matrix}\right))\left(\begin{matrix}6&-5\\-7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\-7&2\end{matrix}\right))\left(\begin{matrix}-36\\39\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\-7&2\end{matrix}\right))\left(\begin{matrix}-36\\39\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\-7&2\end{matrix}\right))\left(\begin{matrix}-36\\39\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{6\times 2-\left(-5\left(-7\right)\right)}&-\frac{-5}{6\times 2-\left(-5\left(-7\right)\right)}\\-\frac{-7}{6\times 2-\left(-5\left(-7\right)\right)}&\frac{6}{6\times 2-\left(-5\left(-7\right)\right)}\end{matrix}\right)\left(\begin{matrix}-36\\39\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{23}&-\frac{5}{23}\\-\frac{7}{23}&-\frac{6}{23}\end{matrix}\right)\left(\begin{matrix}-36\\39\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{23}\left(-36\right)-\frac{5}{23}\times 39\\-\frac{7}{23}\left(-36\right)-\frac{6}{23}\times 39\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{123}{23}\\\frac{18}{23}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{123}{23},y=\frac{18}{23}$$

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.

$$6x-5y=-36,-7x+2y=39$$

Kia ōrite ai a 6x me -7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$-7\times 6x-7\left(-5\right)y=-7\left(-36\right),6\left(-7\right)x+6\times 2y=6\times 39$$

Whakarūnātia.

$$-42x+35y=252,-42x+12y=234$$

Me tango -42x+12y=234 mai i -42x+35y=252 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-42x+42x+35y-12y=252-234$$

Tāpiri -42x ki te 42x. Ka whakakore atu ngā kupu -42x me 42x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$35y-12y=252-234$$

Tāpiri 35y ki te -12y.

$$23y=252-234$$

Tāpiri 252 ki te -234.

$$23y=18$$

Whakawehea ngā taha e rua ki te 23.

$$y=\frac{18}{23}$$

Whakaurua te \frac{18}{23} mō y ki -7x+2y=39. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-7x+2\times \frac{18}{23}=39$$

Whakareatia 2 ki te \frac{18}{23}.

$$-7x+\frac{36}{23}=39$$

Me tango \frac{36}{23} mai i ngā taha e rua o te whārite.

$$-7x=\frac{861}{23}$$

Whakawehea ngā taha e rua ki te -7.

$$x=-\frac{123}{23}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{123}{23},y=\frac{18}{23}$$