Whakaoti i te {l}{-5x+3y=3}{4x+3y=30}

$$\left. \begin{array} { l } { - 5 x + 3 y = 3 } \\ { 4 x + 3 y = 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=6$$

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.

$$-5x+3y=3,4x+3y=30$$

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

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

Me tango 3y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te -5.

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

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

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

Whakakapia te \frac{-3+3y}{5} mō te x ki tērā atu whārite, 4x+3y=30.

$$4\left(\frac{3}{5}y-\frac{3}{5}\right)+3y=30$$

Whakareatia 4 ki te \frac{-3+3y}{5}.

$$\frac{12}{5}y-\frac{12}{5}+3y=30$$

Tāpiri \frac{12y}{5} ki te 3y.

$$\frac{27}{5}y-\frac{12}{5}=30$$

Me tāpiri \frac{12}{5} ki ngā taha e rua o te whārite.

$$\frac{27}{5}y=\frac{162}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{27}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=6$$

Whakaurua te 6 mō y ki x=\frac{3}{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.

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

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

$$x=\frac{18-3}{5}$$

Tāpiri -\frac{3}{5} ki te \frac{18}{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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=6$$

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.

$$-5x+3y=3,4x+3y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&3\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&3\\4&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&3\\4&3\end{matrix}\right))\left(\begin{matrix}-5&3\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&3\\4&3\end{matrix}\right))\left(\begin{matrix}3\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&3\\4&3\end{matrix}\right))\left(\begin{matrix}3\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&3\\4&3\end{matrix}\right))\left(\begin{matrix}3\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{-5\times 3-3\times 4}&-\frac{3}{-5\times 3-3\times 4}\\-\frac{4}{-5\times 3-3\times 4}&-\frac{5}{-5\times 3-3\times 4}\end{matrix}\right)\left(\begin{matrix}3\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{9}&\frac{1}{9}\\\frac{4}{27}&\frac{5}{27}\end{matrix}\right)\left(\begin{matrix}3\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{9}\times 3+\frac{1}{9}\times 30\\\frac{4}{27}\times 3+\frac{5}{27}\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=6$$

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.

$$-5x+3y=3,4x+3y=30$$

Me tango 4x+3y=30 mai i -5x+3y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5x-4x+3y-3y=3-30$$

Tāpiri 3y ki te -3y. Ka whakakore atu ngā kupu 3y me -3y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-5x-4x=3-30$$

Tāpiri -5x ki te -4x.

$$-9x=3-30$$

Tāpiri 3 ki te -30.

$$-9x=-27$$

Whakawehea ngā taha e rua ki te -9.

$$x=3$$

Whakaurua te 3 mō x ki 4x+3y=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$4\times 3+3y=30$$

Whakareatia 4 ki te 3.

$$12+3y=30$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$3y=18$$

Whakawehea ngā taha e rua ki te 3.

$$y=6$$

Kua oti te pūnaha te whakatau.

$$x=3,y=6$$