Whakaoti i te {l}{5x-6y=-3}{5x-3y=3}

$$\left\{ \begin{array} { l } { 5 x - 6 y = - 3 } \\ { 5 x - 3 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{9}{5} = 1\frac{4}{5} = 1.8$$

$$y=2$$

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.

$$5x-6y=-3,5x-3y=3$$

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

$$5x-6y=-3$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$5x=6y-3$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(6y-3\right)$$

Whakareatia \frac{1}{5} ki te 6y-3.

$$x=\frac{6}{5}y-\frac{3}{5}$$

Whakakapia te \frac{6y-3}{5} mō te x ki tērā atu whārite, 5x-3y=3.

$$5\left(\frac{6}{5}y-\frac{3}{5}\right)-3y=3$$

Whakareatia 5 ki te \frac{6y-3}{5}.

$$6y-3-3y=3$$

Tāpiri 6y ki te -3y.

$$3y-3=3$$

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

$$3y=6$$

Whakawehea ngā taha e rua ki te 3.

$$y=2$$

Whakaurua te 2 mō y ki x=\frac{6}{5}y-\frac{3}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{6}{5}\times 2-\frac{3}{5}$$

Whakareatia \frac{6}{5} ki te 2.

$$x=\frac{12-3}{5}$$

Tāpiri -\frac{3}{5} ki te \frac{12}{5} 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.

$$x=\frac{9}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{5},y=2$$

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.

$$5x-6y=-3,5x-3y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-6\\5&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-6\\5&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-6\\5&-3\end{matrix}\right))\left(\begin{matrix}5&-6\\5&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\5&-3\end{matrix}\right))\left(\begin{matrix}-3\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\5&-3\end{matrix}\right))\left(\begin{matrix}-3\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\5&-3\end{matrix}\right))\left(\begin{matrix}-3\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5\left(-3\right)-\left(-6\times 5\right)}&-\frac{-6}{5\left(-3\right)-\left(-6\times 5\right)}\\-\frac{5}{5\left(-3\right)-\left(-6\times 5\right)}&\frac{5}{5\left(-3\right)-\left(-6\times 5\right)}\end{matrix}\right)\left(\begin{matrix}-3\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{5}\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{9}{5},y=2$$

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.

$$5x-6y=-3,5x-3y=3$$

Me tango 5x-3y=3 mai i 5x-6y=-3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y+3y=-3-3$$

Tāpiri -6y ki te 3y.

$$-3y=-3-3$$

Tāpiri -3 ki te -3.

$$-3y=-6$$

Whakawehea ngā taha e rua ki te -3.

$$y=2$$

Whakaurua te 2 mō y ki 5x-3y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-3\times 2=3$$

Whakareatia -3 ki te 2.

$$5x-6=3$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$5x=9$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{9}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{5},y=2$$