Whakaoti i te {l}{2x+5y+1=0}{3x-2y-8=0}

$$\left\{ \begin{array} { l } { 2 x + 5 y + 1 = 0 } \\ { 3 x - 2 y - 8 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-1$$

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.

$$2x+5y+1=0,3x-2y-8=0$$

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

$$2x+5y+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$2x+5y=-1$$

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

$$2x=-5y-1$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y-1\right)$$

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

$$x=-\frac{5}{2}y-\frac{1}{2}$$

Whakakapia te \frac{-5y-1}{2} mō te x ki tērā atu whārite, 3x-2y-8=0.

$$3\left(-\frac{5}{2}y-\frac{1}{2}\right)-2y-8=0$$

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

$$-\frac{15}{2}y-\frac{3}{2}-2y-8=0$$

Tāpiri -\frac{15y}{2} ki te -2y.

$$-\frac{19}{2}y-\frac{3}{2}-8=0$$

Tāpiri -\frac{3}{2} ki te -8.

$$-\frac{19}{2}y-\frac{19}{2}=0$$

Me tāpiri \frac{19}{2} ki ngā taha e rua o te whārite.

$$-\frac{19}{2}y=\frac{19}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{19}{2}, 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{5}{2}y-\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}\left(-1\right)-\frac{1}{2}$$

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

$$x=\frac{5-1}{2}$$

Tāpiri -\frac{1}{2} ki te \frac{5}{2} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-1$$

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.

$$2x+5y+1=0,3x-2y-8=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=-1$$

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.

$$2x+5y+1=0,3x-2y-8=0$$

Kia ōrite ai a 2x 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 2.

$$3\times 2x+3\times 5y+3=0,2\times 3x+2\left(-2\right)y+2\left(-8\right)=0$$

Whakarūnātia.

$$6x+15y+3=0,6x-4y-16=0$$

Me tango 6x-4y-16=0 mai i 6x+15y+3=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+15y+4y+3+16=0$$

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.

$$15y+4y+3+16=0$$

Tāpiri 15y ki te 4y.

$$19y+3+16=0$$

Tāpiri 3 ki te 16.

$$19y+19=0$$

Me tango 19 mai i ngā taha e rua o te whārite.

$$19y=-19$$

Whakawehea ngā taha e rua ki te 19.

$$y=-1$$

Whakaurua te -1 mō y ki 3x-2y-8=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-2\left(-1\right)-8=0$$

Whakareatia -2 ki te -1.

$$3x+2-8=0$$

Tāpiri 2 ki te -8.

$$3x-6=0$$

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

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-1$$