Whakaoti i te {l}{-5x+y=-3}{3x-8y=24}

$$\left. \begin{array} { l } { - 5 x + y = - 3 } \\ { 3 x - 8 y = 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=-3$$

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.

$$-5x+y=-3,3x-8y=24$$

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

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

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

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

Whakawehea ngā taha e rua ki te -5.

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

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

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

Whakakapia te \frac{3+y}{5} mō te x ki tērā atu whārite, 3x-8y=24.

$$3\left(\frac{1}{5}y+\frac{3}{5}\right)-8y=24$$

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

$$\frac{3}{5}y+\frac{9}{5}-8y=24$$

Tāpiri \frac{3y}{5} ki te -8y.

$$-\frac{37}{5}y+\frac{9}{5}=24$$

Me tango \frac{9}{5} mai i ngā taha e rua o te whārite.

$$-\frac{37}{5}y=\frac{111}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{37}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-3$$

Whakaurua te -3 mō y ki x=\frac{1}{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.

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

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

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

Tāpiri \frac{3}{5} ki te -\frac{3}{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.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-3$$

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.

$$-5x+y=-3,3x-8y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&1\\3&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&1\\3&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&1\\3&-8\end{matrix}\right))\left(\begin{matrix}-5&1\\3&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&1\\3&-8\end{matrix}\right))\left(\begin{matrix}-3\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&1\\3&-8\end{matrix}\right))\left(\begin{matrix}-3\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&1\\3&-8\end{matrix}\right))\left(\begin{matrix}-3\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{-5\left(-8\right)-3}&-\frac{1}{-5\left(-8\right)-3}\\-\frac{3}{-5\left(-8\right)-3}&-\frac{5}{-5\left(-8\right)-3}\end{matrix}\right)\left(\begin{matrix}-3\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{37}&-\frac{1}{37}\\-\frac{3}{37}&-\frac{5}{37}\end{matrix}\right)\left(\begin{matrix}-3\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{37}\left(-3\right)-\frac{1}{37}\times 24\\-\frac{3}{37}\left(-3\right)-\frac{5}{37}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-3$$

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.

$$-5x+y=-3,3x-8y=24$$

Kia ōrite ai a -5x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -5.

$$3\left(-5\right)x+3y=3\left(-3\right),-5\times 3x-5\left(-8\right)y=-5\times 24$$

Whakarūnātia.

$$-15x+3y=-9,-15x+40y=-120$$

Me tango -15x+40y=-120 mai i -15x+3y=-9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-15x+15x+3y-40y=-9+120$$

Tāpiri -15x ki te 15x. Ka whakakore atu ngā kupu -15x me 15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-40y=-9+120$$

Tāpiri 3y ki te -40y.

$$-37y=-9+120$$

Tāpiri -9 ki te 120.

$$-37y=111$$

Whakawehea ngā taha e rua ki te -37.

$$y=-3$$

Whakaurua te -3 mō y ki 3x-8y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-8\left(-3\right)=24$$

Whakareatia -8 ki te -3.

$$3x+24=24$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$3x=0$$

Whakawehea ngā taha e rua ki te 3.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-3$$