Whakaoti i te {l}{3x-4y=7}{x+3/2-y=4}

$$\left\{ \begin{array} { l } { 3 x - 4 y = 7 } \\ { \frac { x + 3 } { 2 } - y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=-4$$

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.

$$3x-4y=7,\frac{1}{2}\left(x+3\right)-y=4$$

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

$$3x-4y=7$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$3x=4y+7$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(4y+7\right)$$

Whakareatia \frac{1}{3} ki te 4y+7.

$$x=\frac{4}{3}y+\frac{7}{3}$$

Whakakapia te \frac{4y+7}{3} mō te x ki tērā atu whārite, \frac{1}{2}\left(x+3\right)-y=4.

$$\frac{1}{2}\left(\frac{4}{3}y+\frac{7}{3}+3\right)-y=4$$

Tāpiri \frac{7}{3} ki te 3.

$$\frac{1}{2}\left(\frac{4}{3}y+\frac{16}{3}\right)-y=4$$

Whakareatia \frac{1}{2} ki te \frac{16+4y}{3}.

$$\frac{2}{3}y+\frac{8}{3}-y=4$$

Tāpiri \frac{2y}{3} ki te -y.

$$-\frac{1}{3}y+\frac{8}{3}=4$$

Me tango \frac{8}{3} mai i ngā taha e rua o te whārite.

$$-\frac{1}{3}y=\frac{4}{3}$$

Me whakarea ngā taha e rua ki te -3.

$$y=-4$$

Whakaurua te -4 mō y ki x=\frac{4}{3}y+\frac{7}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4}{3}\left(-4\right)+\frac{7}{3}$$

Whakareatia \frac{4}{3} ki te -4.

$$x=\frac{-16+7}{3}$$

Tāpiri \frac{7}{3} ki te -\frac{16}{3} 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.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-4$$

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.

$$3x-4y=7,\frac{1}{2}\left(x+3\right)-y=4$$

Whakarūnātia te whārite tuarua ki te āhua tānga ngahuru.

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

Whakareatia \frac{1}{2} ki te x+3.

$$\frac{1}{2}x+\frac{3}{2}-y=4$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-4\\\frac{1}{2}&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\\frac{5}{2}\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-4\\\frac{1}{2}&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-4\\\frac{1}{2}&-1\end{matrix}\right))\left(\begin{matrix}3&-4\\\frac{1}{2}&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\\frac{1}{2}&-1\end{matrix}\right))\left(\begin{matrix}7\\\frac{5}{2}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\\frac{1}{2}&-1\end{matrix}\right))\left(\begin{matrix}7\\\frac{5}{2}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\\frac{1}{2}&-1\end{matrix}\right))\left(\begin{matrix}7\\\frac{5}{2}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-\left(-4\times \frac{1}{2}\right)}&-\frac{-4}{3\left(-1\right)-\left(-4\times \frac{1}{2}\right)}\\-\frac{\frac{1}{2}}{3\left(-1\right)-\left(-4\times \frac{1}{2}\right)}&\frac{3}{3\left(-1\right)-\left(-4\times \frac{1}{2}\right)}\end{matrix}\right)\left(\begin{matrix}7\\\frac{5}{2}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&-4\\\frac{1}{2}&-3\end{matrix}\right)\left(\begin{matrix}7\\\frac{5}{2}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7-4\times \frac{5}{2}\\\frac{1}{2}\times 7-3\times \frac{5}{2}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=-4$$