Whakaoti i te {l}{x-1/4+y+2/4=5/12}{6x-y=1}

$$\left\{ \begin{array} { l } { \frac { x - 1 } { 4 } + \frac { y + 2 } { 4 } = \frac { 5 } { 12 } } \\ { 6 x - y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{5}{21}\approx 0.238095238$$

$$y=\frac{3}{7}\approx 0.428571429$$

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.

$$\frac{1}{4}\left(x-1\right)+\frac{1}{4}\left(y+2\right)=\frac{5}{12},6x-y=1$$

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

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

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

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

Whakareatia \frac{1}{4} ki te y+2.

$$\frac{1}{4}x-\frac{1}{4}+\frac{1}{4}y+\frac{1}{2}=\frac{5}{12}$$

Tāpiri -\frac{1}{4} ki te \frac{1}{2} 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.

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

Me tango \frac{1}{4} mai i ngā taha e rua o te whārite.

$$\frac{1}{4}x+\frac{1}{4}y=\frac{1}{6}$$

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

$$\frac{1}{4}x=-\frac{1}{4}y+\frac{1}{6}$$

Me whakarea ngā taha e rua ki te 4.

$$x=4\left(-\frac{1}{4}y+\frac{1}{6}\right)$$

Whakareatia 4 ki te -\frac{y}{4}+\frac{1}{6}.

$$x=-y+\frac{2}{3}$$

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

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

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

$$-6y+4-y=1$$

Tāpiri -6y ki te -y.

$$-7y+4=1$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-7y=-3$$

Whakawehea ngā taha e rua ki te -7.

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

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

$$x=-\frac{3}{7}+\frac{2}{3}$$

Tāpiri \frac{2}{3} ki te -\frac{3}{7} 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.

$$x=\frac{5}{21}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{21},y=\frac{3}{7}$$

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.

$$\frac{1}{4}\left(x-1\right)+\frac{1}{4}\left(y+2\right)=\frac{5}{12},6x-y=1$$

Whakarūnātia te whārite tuatahi ki te āhua tānga ngahuru.

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

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

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

Whakareatia \frac{1}{4} ki te y+2.

$$\frac{1}{4}x-\frac{1}{4}+\frac{1}{4}y+\frac{1}{2}=\frac{5}{12}$$

Tāpiri -\frac{1}{4} ki te \frac{1}{2} 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.

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

Me tango \frac{1}{4} mai i ngā taha e rua o te whārite.

$$\frac{1}{4}x+\frac{1}{4}y=\frac{1}{6}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\6&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\6&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\6&-1\end{matrix}\right))\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\6&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\6&-1\end{matrix}\right))\left(\begin{matrix}\frac{1}{6}\\1\end{matrix}\right)$$

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

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

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\6&-1\end{matrix}\right))\left(\begin{matrix}\frac{1}{6}\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{\frac{1}{4}\left(-1\right)-\frac{1}{4}\times 6}&-\frac{\frac{1}{4}}{\frac{1}{4}\left(-1\right)-\frac{1}{4}\times 6}\\-\frac{6}{\frac{1}{4}\left(-1\right)-\frac{1}{4}\times 6}&\frac{\frac{1}{4}}{\frac{1}{4}\left(-1\right)-\frac{1}{4}\times 6}\end{matrix}\right)\left(\begin{matrix}\frac{1}{6}\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7}&\frac{1}{7}\\\frac{24}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}\frac{1}{6}\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7}\times \frac{1}{6}+\frac{1}{7}\\\frac{24}{7}\times \frac{1}{6}-\frac{1}{7}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}\\\frac{3}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{21},y=\frac{3}{7}$$