Whakaoti i te {l}{3x+y=5}{2x+y=10}

$$\left. \begin{array} { l } { 3 x + y = 5 } \\ { 2 x + y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-5$$

$$y=20$$

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.

$$3x+y=5,2x+y=10$$

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

$$3x+y=5$$

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

$$3x=-y+5$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=-\frac{1}{3}y+\frac{5}{3}$$

Whakakapia te \frac{-y+5}{3} mō te x ki tērā atu whārite, 2x+y=10.

$$2\left(-\frac{1}{3}y+\frac{5}{3}\right)+y=10$$

Whakareatia 2 ki te \frac{-y+5}{3}.

$$-\frac{2}{3}y+\frac{10}{3}+y=10$$

Tāpiri -\frac{2y}{3} ki te y.

$$\frac{1}{3}y+\frac{10}{3}=10$$

Me tango \frac{10}{3} mai i ngā taha e rua o te whārite.

$$\frac{1}{3}y=\frac{20}{3}$$

Me whakarea ngā taha e rua ki te 3.

$$y=20$$

Whakaurua te 20 mō y ki x=-\frac{1}{3}y+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times 20+\frac{5}{3}$$

Whakareatia -\frac{1}{3} ki te 20.

$$x=\frac{-20+5}{3}$$

Tāpiri \frac{5}{3} ki te -\frac{20}{3} 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.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=20$$

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.

$$3x+y=5,2x+y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&1\\2&1\end{matrix}\right))\left(\begin{matrix}3&1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&1\end{matrix}\right))\left(\begin{matrix}5\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&1\end{matrix}\right))\left(\begin{matrix}5\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&1\end{matrix}\right))\left(\begin{matrix}5\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-2}&-\frac{1}{3-2}\\-\frac{2}{3-2}&\frac{3}{3-2}\end{matrix}\right)\left(\begin{matrix}5\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&-1\\-2&3\end{matrix}\right)\left(\begin{matrix}5\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5-10\\-2\times 5+3\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\20\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-5,y=20$$

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.

$$3x+y=5,2x+y=10$$

Me tango 2x+y=10 mai i 3x+y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-2x+y-y=5-10$$

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

$$3x-2x=5-10$$

Tāpiri 3x ki te -2x.

$$x=5-10$$

Tāpiri 5 ki te -10.

$$x=-5$$

Whakaurua te -5 mō x ki 2x+y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2\left(-5\right)+y=10$$

Whakareatia 2 ki te -5.

$$-10+y=10$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$y=20$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=20$$