Whakaoti i te {l}{3y-7x=-9}{5x+2y=23}

$$\left\{ \begin{array} { l } { 3 y - 7 x = - 9 } \\ { 5 x + 2 y = 23 } \end{array} \right.$$

Whakaoti mō y, x

$$x=3$$

$$y=4$$

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.

$$3y-7x=-9,2y+5x=23$$

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

$$3y-7x=-9$$

Me tāpiri 7x ki ngā taha e rua o te whārite.

$$3y=7x-9$$

Whakawehea ngā taha e rua ki te 3.

$$y=\frac{1}{3}\left(7x-9\right)$$

Whakareatia \frac{1}{3} ki te 7x-9.

$$y=\frac{7}{3}x-3$$

Whakakapia te \frac{7x}{3}-3 mō te y ki tērā atu whārite, 2y+5x=23.

$$2\left(\frac{7}{3}x-3\right)+5x=23$$

Whakareatia 2 ki te \frac{7x}{3}-3.

$$\frac{14}{3}x-6+5x=23$$

Tāpiri \frac{14x}{3} ki te 5x.

$$\frac{29}{3}x-6=23$$

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

$$\frac{29}{3}x=29$$

Whakawehea ngā taha e rua o te whārite ki te \frac{29}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=3$$

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

$$y=\frac{7}{3}\times 3-3$$

Whakareatia \frac{7}{3} ki te 3.

$$y=7-3$$

Tāpiri -3 ki te 7.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$y=4,x=3$$

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.

$$3y-7x=-9,2y+5x=23$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-7\\2&5\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-9\\23\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-7\\2&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-7\\2&5\end{matrix}\right))\left(\begin{matrix}3&-7\\2&5\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-7\\2&5\end{matrix}\right))\left(\begin{matrix}-9\\23\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-7\\2&5\end{matrix}\right))\left(\begin{matrix}-9\\23\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-7\\2&5\end{matrix}\right))\left(\begin{matrix}-9\\23\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3\times 5-\left(-7\times 2\right)}&-\frac{-7}{3\times 5-\left(-7\times 2\right)}\\-\frac{2}{3\times 5-\left(-7\times 2\right)}&\frac{3}{3\times 5-\left(-7\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-9\\23\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{29}&\frac{7}{29}\\-\frac{2}{29}&\frac{3}{29}\end{matrix}\right)\left(\begin{matrix}-9\\23\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{29}\left(-9\right)+\frac{7}{29}\times 23\\-\frac{2}{29}\left(-9\right)+\frac{3}{29}\times 23\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=4,x=3$$

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.

$$3y-7x=-9,2y+5x=23$$

Kia ōrite ai a 3y me 2y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3y+2\left(-7\right)x=2\left(-9\right),3\times 2y+3\times 5x=3\times 23$$

Whakarūnātia.

$$6y-14x=-18,6y+15x=69$$

Me tango 6y+15x=69 mai i 6y-14x=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6y-6y-14x-15x=-18-69$$

Tāpiri 6y ki te -6y. Ka whakakore atu ngā kupu 6y me -6y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-14x-15x=-18-69$$

Tāpiri -14x ki te -15x.

$$-29x=-18-69$$

Tāpiri -18 ki te -69.

$$-29x=-87$$

Whakawehea ngā taha e rua ki te -29.

$$x=3$$

Whakaurua te 3 mō x ki 2y+5x=23. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y+5\times 3=23$$

Whakareatia 5 ki te 3.

$$2y+15=23$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$2y=8$$

Whakawehea ngā taha e rua ki te 2.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$y=4,x=3$$