Whakaoti i te {l}{2x-7y=8}{y-2x=-3.20}

$$\left\{ \begin{array} { l } { 2 x - 7 y = 8 } \\ { y - 2 x = - 3.20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1.2$$

$$y=-0.8$$

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.

$$2x-7y=8,-2x+y=-3.2$$

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

$$2x-7y=8$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$2x=7y+8$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(7y+8\right)$$

Whakareatia \frac{1}{2} ki te 7y+8.

$$x=\frac{7}{2}y+4$$

Whakakapia te \frac{7y}{2}+4 mō te x ki tērā atu whārite, -2x+y=-3.2.

$$-2\left(\frac{7}{2}y+4\right)+y=-3.2$$

Whakareatia -2 ki te \frac{7y}{2}+4.

$$-7y-8+y=-3.2$$

Tāpiri -7y ki te y.

$$-6y-8=-3.2$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$-6y=4.8$$

Whakawehea ngā taha e rua ki te -6.

$$y=-0.8$$

Whakaurua te -0.8 mō y ki x=\frac{7}{2}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{2}\left(-0.8\right)+4$$

Whakareatia \frac{7}{2} ki te -0.8 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.

$$x=-\frac{14}{5}+4$$

Tāpiri 4 ki te -\frac{14}{5}.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{6}{5},y=-0.8$$

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.

$$2x-7y=8,-2x+y=-3.2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-7\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\-3.2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-7\\-2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-7\\-2&1\end{matrix}\right))\left(\begin{matrix}2&-7\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-7\\-2&1\end{matrix}\right))\left(\begin{matrix}8\\-3.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-7\\-2&1\end{matrix}\right))\left(\begin{matrix}8\\-3.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-7\\-2&1\end{matrix}\right))\left(\begin{matrix}8\\-3.2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-7\left(-2\right)\right)}&-\frac{-7}{2-\left(-7\left(-2\right)\right)}\\-\frac{-2}{2-\left(-7\left(-2\right)\right)}&\frac{2}{2-\left(-7\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}8\\-3.2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}&-\frac{7}{12}\\-\frac{1}{6}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}8\\-3.2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}\times 8-\frac{7}{12}\left(-3.2\right)\\-\frac{1}{6}\times 8-\frac{1}{6}\left(-3.2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5}\\-\frac{4}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{6}{5},y=-\frac{4}{5}$$

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.

$$2x-7y=8,-2x+y=-3.2$$

Kia ōrite ai a 2x me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-2\times 2x-2\left(-7\right)y=-2\times 8,2\left(-2\right)x+2y=2\left(-3.2\right)$$

Whakarūnātia.

$$-4x+14y=-16,-4x+2y=-6.4$$

Me tango -4x+2y=-6.4 mai i -4x+14y=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x+14y-2y=-16+6.4$$

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$14y-2y=-16+6.4$$

Tāpiri 14y ki te -2y.

$$12y=-16+6.4$$

Tāpiri -16 ki te 6.4.

$$12y=-9.6$$

Whakawehea ngā taha e rua ki te 12.

$$y=-\frac{4}{5}$$

Whakaurua te -\frac{4}{5} mō y ki -2x+y=-3.2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-\frac{4}{5}=-3.2$$

Me tāpiri \frac{4}{5} ki ngā taha e rua o te whārite.

$$-2x=-\frac{12}{5}$$

Whakawehea ngā taha e rua ki te -2.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{6}{5},y=-\frac{4}{5}$$