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

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

Whakaoti mō x, y

$$x=\frac{5}{33}\approx 0.151515152$$

$$y=-\frac{17}{33}\approx -0.515151515$$

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.

$$3x-3y=2,4x+7y=-3$$

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

$$3x-3y=2$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$3x=3y+2$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

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

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

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

$$4y+\frac{8}{3}+7y=-3$$

Tāpiri 4y ki te 7y.

$$11y+\frac{8}{3}=-3$$

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

$$11y=-\frac{17}{3}$$

Whakawehea ngā taha e rua ki te 11.

$$y=-\frac{17}{33}$$

Whakaurua te -\frac{17}{33} 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.

$$x=-\frac{17}{33}+\frac{2}{3}$$

Tāpiri \frac{2}{3} ki te -\frac{17}{33} 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.

$$x=\frac{5}{33}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{33},y=-\frac{17}{33}$$

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.

$$3x-3y=2,4x+7y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-3\\4&7\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{33}&\frac{1}{11}\\-\frac{4}{33}&\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}2\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{33}\times 2+\frac{1}{11}\left(-3\right)\\-\frac{4}{33}\times 2+\frac{1}{11}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{33}\\-\frac{17}{33}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{33},y=-\frac{17}{33}$$

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.

$$3x-3y=2,4x+7y=-3$$

Kia ōrite ai a 3x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$4\times 3x+4\left(-3\right)y=4\times 2,3\times 4x+3\times 7y=3\left(-3\right)$$

Whakarūnātia.

$$12x-12y=8,12x+21y=-9$$

Me tango 12x+21y=-9 mai i 12x-12y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-12y-21y=8+9$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12y-21y=8+9$$

Tāpiri -12y ki te -21y.

$$-33y=8+9$$

Tāpiri 8 ki te 9.

$$-33y=17$$

Whakawehea ngā taha e rua ki te -33.

$$y=-\frac{17}{33}$$

Whakaurua te -\frac{17}{33} mō y ki 4x+7y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+7\left(-\frac{17}{33}\right)=-3$$

Whakareatia 7 ki te -\frac{17}{33}.

$$4x-\frac{119}{33}=-3$$

Me tāpiri \frac{119}{33} ki ngā taha e rua o te whārite.

$$4x=\frac{20}{33}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{5}{33}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{33},y=-\frac{17}{33}$$