Whakaoti i te {l}{3(2x+1)-5(y-3)=1}{5(1-x)-4(2y+1)=3}

$$\left\{ \begin{array} { l } { 3 ( 2 x + 1 ) - 5 ( y - 3 ) = 1 } \\ { 5 ( 1 - x ) - 4 ( 2 y + 1 ) = 3 } \end{array} \right.$$

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.

$$3\left(2x+1\right)-5\left(y-3\right)=1,5\left(-x+1\right)-4\left(2y+1\right)=3$$

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

$$3\left(2x+1\right)-5\left(y-3\right)=1$$

Whakareatia 3 ki te 2x+1.

$$6x+3-5\left(y-3\right)=1$$

Whakareatia -5 ki te y-3.

$$6x+3-5y+15=1$$

Tāpiri 3 ki te 15.

$$6x-5y+18=1$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$6x-5y=-17$$

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

$$6x=5y-17$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(5y-17\right)$$

Whakareatia \frac{1}{6} ki te 5y-17.

$$x=\frac{5}{6}y-\frac{17}{6}$$

Whakakapia te \frac{5y-17}{6} mō te x ki tērā atu whārite, 5\left(-x+1\right)-4\left(2y+1\right)=3.

$$5\left(-\left(\frac{5}{6}y-\frac{17}{6}\right)+1\right)-4\left(2y+1\right)=3$$

Whakareatia -1 ki te \frac{5y-17}{6}.

$$5\left(-\frac{5}{6}y+\frac{17}{6}+1\right)-4\left(2y+1\right)=3$$

Tāpiri \frac{17}{6} ki te 1.

$$5\left(-\frac{5}{6}y+\frac{23}{6}\right)-4\left(2y+1\right)=3$$

Whakareatia 5 ki te \frac{-5y+23}{6}.

$$-\frac{25}{6}y+\frac{115}{6}-4\left(2y+1\right)=3$$

Whakareatia -4 ki te 2y+1.

$$-\frac{25}{6}y+\frac{115}{6}-8y-4=3$$

Tāpiri -\frac{25y}{6} ki te -8y.

$$-\frac{73}{6}y+\frac{115}{6}-4=3$$

Tāpiri \frac{115}{6} ki te -4.

$$-\frac{73}{6}y+\frac{91}{6}=3$$

Me tango \frac{91}{6} mai i ngā taha e rua o te whārite.

$$-\frac{73}{6}y=-\frac{73}{6}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{73}{6}, 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{5}{6}y-\frac{17}{6}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5-17}{6}$$

Tāpiri -\frac{17}{6} ki te \frac{5}{6} 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.

$$3\left(2x+1\right)-5\left(y-3\right)=1,5\left(-x+1\right)-4\left(2y+1\right)=3$$

Whakarūnātia te whārite tuatahi ki te āhua tānga ngahuru.

$$3\left(2x+1\right)-5\left(y-3\right)=1$$

Whakareatia 3 ki te 2x+1.

$$6x+3-5\left(y-3\right)=1$$

Whakareatia -5 ki te y-3.

$$6x+3-5y+15=1$$

Tāpiri 3 ki te 15.

$$6x-5y+18=1$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$6x-5y=-17$$

Whakarūnātia te whārite tuarua ki te āhua tānga ngahuru.

$$5\left(-x+1\right)-4\left(2y+1\right)=3$$

Whakareatia 5 ki te -x+1.

$$-5x+5-4\left(2y+1\right)=3$$

Whakareatia -4 ki te 2y+1.

$$-5x+5-8y-4=3$$

Tāpiri 5 ki te -4.

$$-5x-8y+1=3$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-5x-8y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&-5\\-5&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-17\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&-5\\-5&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&-5\\-5&-8\end{matrix}\right))\left(\begin{matrix}6&-5\\-5&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\-5&-8\end{matrix}\right))\left(\begin{matrix}-17\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\-5&-8\end{matrix}\right))\left(\begin{matrix}-17\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\-5&-8\end{matrix}\right))\left(\begin{matrix}-17\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{6\left(-8\right)-\left(-5\left(-5\right)\right)}&-\frac{-5}{6\left(-8\right)-\left(-5\left(-5\right)\right)}\\-\frac{-5}{6\left(-8\right)-\left(-5\left(-5\right)\right)}&\frac{6}{6\left(-8\right)-\left(-5\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}-17\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{73}&-\frac{5}{73}\\-\frac{5}{73}&-\frac{6}{73}\end{matrix}\right)\left(\begin{matrix}-17\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{73}\left(-17\right)-\frac{5}{73}\times 2\\-\frac{5}{73}\left(-17\right)-\frac{6}{73}\times 2\end{matrix}\right)$$

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.

$$x=-2,y=1$$