Whakaoti i te {c}{3x-2y=1}{-5x-3y=-8}

$$\left\{ \begin{array} { c } { 3 x - 2 y = 1 } \\ { - 5 x - 3 y = - 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=1$$

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.

$$3x-2y=1,-5x-3y=-8$$

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

$$3x-2y=1$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$3x=2y+1$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(2y+1\right)$$

Whakareatia \frac{1}{3} ki te 2y+1.

$$x=\frac{2}{3}y+\frac{1}{3}$$

Whakakapia te \frac{2y+1}{3} mō te x ki tērā atu whārite, -5x-3y=-8.

$$-5\left(\frac{2}{3}y+\frac{1}{3}\right)-3y=-8$$

Whakareatia -5 ki te \frac{2y+1}{3}.

$$-\frac{10}{3}y-\frac{5}{3}-3y=-8$$

Tāpiri -\frac{10y}{3} ki te -3y.

$$-\frac{19}{3}y-\frac{5}{3}=-8$$

Me tāpiri \frac{5}{3} ki ngā taha e rua o te whārite.

$$-\frac{19}{3}y=-\frac{19}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{19}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{2}{3}y+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2+1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{2}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$

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.

$$3x-2y=1,-5x-3y=-8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\-5&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-2\\-5&-3\end{matrix}\right))\left(\begin{matrix}3&-2\\-5&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-5&-3\end{matrix}\right))\left(\begin{matrix}1\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-5&-3\end{matrix}\right))\left(\begin{matrix}1\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-5&-3\end{matrix}\right))\left(\begin{matrix}1\\-8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{3\left(-3\right)-\left(-2\left(-5\right)\right)}&-\frac{-2}{3\left(-3\right)-\left(-2\left(-5\right)\right)}\\-\frac{-5}{3\left(-3\right)-\left(-2\left(-5\right)\right)}&\frac{3}{3\left(-3\right)-\left(-2\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}1\\-8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{19}&-\frac{2}{19}\\-\frac{5}{19}&-\frac{3}{19}\end{matrix}\right)\left(\begin{matrix}1\\-8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{19}-\frac{2}{19}\left(-8\right)\\-\frac{5}{19}-\frac{3}{19}\left(-8\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=1$$

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.

$$3x-2y=1,-5x-3y=-8$$

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

$$-5\times 3x-5\left(-2\right)y=-5,3\left(-5\right)x+3\left(-3\right)y=3\left(-8\right)$$

Whakarūnātia.

$$-15x+10y=-5,-15x-9y=-24$$

Me tango -15x-9y=-24 mai i -15x+10y=-5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-15x+15x+10y+9y=-5+24$$

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.

$$10y+9y=-5+24$$

Tāpiri 10y ki te 9y.

$$19y=-5+24$$

Tāpiri -5 ki te 24.

$$19y=19$$

Whakawehea ngā taha e rua ki te 19.

$$y=1$$

Whakaurua te 1 mō y ki -5x-3y=-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x-3=-8$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$-5x=-5$$

Whakawehea ngā taha e rua ki te -5.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$