Whakaoti i te {c}{3x+y+5=0}{-2x-y+1=0}

$$\left. \begin{array} { c } { 3 x + y + 5 = 0 } \\ { - 2 x - y + 1 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-6$$

$$y=13$$

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.

$$3x+y+5=0,-2x-y+1=0$$

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

$$3x+y+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$3x+y=-5$$

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+1=0.

$$-2\left(-\frac{1}{3}y-\frac{5}{3}\right)-y+1=0$$

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

$$\frac{2}{3}y+\frac{10}{3}-y+1=0$$

Tāpiri \frac{2y}{3} ki te -y.

$$-\frac{1}{3}y+\frac{10}{3}+1=0$$

Tāpiri \frac{10}{3} ki te 1.

$$-\frac{1}{3}y+\frac{13}{3}=0$$

Me tango \frac{13}{3} mai i ngā taha e rua o te whārite.

$$-\frac{1}{3}y=-\frac{13}{3}$$

Me whakarea ngā taha e rua ki te -3.

$$y=13$$

Whakaurua te 13 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.

$$x=-\frac{1}{3}\times 13-\frac{5}{3}$$

Whakareatia -\frac{1}{3} ki te 13.

$$x=\frac{-13-5}{3}$$

Tāpiri -\frac{5}{3} ki te -\frac{13}{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.

$$x=-6$$

Kua oti te pūnaha te whakatau.

$$x=-6,y=13$$

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.

$$3x+y+5=0,-2x-y+1=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\13\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-6,y=13$$

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.

$$3x+y+5=0,-2x-y+1=0$$

Kia ōrite ai a 3x me -2x, 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.

$$-2\times 3x-2y-2\times 5=0,3\left(-2\right)x+3\left(-1\right)y+3=0$$

Whakarūnātia.

$$-6x-2y-10=0,-6x-3y+3=0$$

Me tango -6x-3y+3=0 mai i -6x-2y-10=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-6x+6x-2y+3y-10-3=0$$

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

$$-2y+3y-10-3=0$$

Tāpiri -2y ki te 3y.

$$y-10-3=0$$

Tāpiri -10 ki te -3.

$$y-13=0$$

Me tāpiri 13 ki ngā taha e rua o te whārite.

$$y=13$$

Whakaurua te 13 mō y ki -2x-y+1=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-13+1=0$$

Tāpiri -13 ki te 1.

$$-2x-12=0$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$-2x=12$$

Whakawehea ngā taha e rua ki te -2.

$$x=-6$$

Kua oti te pūnaha te whakatau.

$$x=-6,y=13$$