Whakaoti i te {l}{5x+3y=30}{3x+3y=18}

$$\left. \begin{array} { l } { 5 x + 3 y = 30 } \\ { 3 x + 3 y = 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=0$$

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.

$$5x+3y=30,3x+3y=18$$

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

$$5x+3y=30$$

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

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

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-3y+30\right)$$

Whakareatia \frac{1}{5} ki te -3y+30.

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

Whakakapia te -\frac{3y}{5}+6 mō te x ki tērā atu whārite, 3x+3y=18.

$$3\left(-\frac{3}{5}y+6\right)+3y=18$$

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

$$-\frac{9}{5}y+18+3y=18$$

Tāpiri -\frac{9y}{5} ki te 3y.

$$\frac{6}{5}y+18=18$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$\frac{6}{5}y=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{6}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{3}{5}y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=0$$

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.

$$5x+3y=30,3x+3y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&3\\3&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\18\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&3\\3&3\end{matrix}\right))\left(\begin{matrix}5&3\\3&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\3&3\end{matrix}\right))\left(\begin{matrix}30\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\3&3\end{matrix}\right))\left(\begin{matrix}30\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\3&3\end{matrix}\right))\left(\begin{matrix}30\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-3\times 3}&-\frac{3}{5\times 3-3\times 3}\\-\frac{3}{5\times 3-3\times 3}&\frac{5}{5\times 3-3\times 3}\end{matrix}\right)\left(\begin{matrix}30\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{2}\\-\frac{1}{2}&\frac{5}{6}\end{matrix}\right)\left(\begin{matrix}30\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 30-\frac{1}{2}\times 18\\-\frac{1}{2}\times 30+\frac{5}{6}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=0$$

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.

$$5x+3y=30,3x+3y=18$$

Me tango 3x+3y=18 mai i 5x+3y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-3x+3y-3y=30-18$$

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.

$$5x-3x=30-18$$

Tāpiri 5x ki te -3x.

$$2x=30-18$$

Tāpiri 30 ki te -18.

$$2x=12$$

Whakawehea ngā taha e rua ki te 2.

$$x=6$$

Whakaurua te 6 mō x ki 3x+3y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3\times 6+3y=18$$

Whakareatia 3 ki te 6.

$$18+3y=18$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$3y=0$$

Whakawehea ngā taha e rua ki te 3.

$$y=0$$

Kua oti te pūnaha te whakatau.

$$x=6,y=0$$