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

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

Whakaoti mō x, y

$$x=-3$$

$$y=-7$$

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.

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

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

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

Me tango 2 mai i ngā taha e rua o te whārite.

$$3x-y=-2$$

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

$$3x=y-2$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

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

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

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

$$\frac{5}{3}y-\frac{10}{3}-2y+1=0$$

Tāpiri \frac{5y}{3} ki te -2y.

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

Tāpiri -\frac{10}{3} ki te 1.

$$-\frac{1}{3}y-\frac{7}{3}=0$$

Me tāpiri \frac{7}{3} ki ngā taha e rua o te whārite.

$$-\frac{1}{3}y=\frac{7}{3}$$

Me whakarea ngā taha e rua ki te -3.

$$y=-7$$

Whakaurua te -7 mō y ki x=\frac{1}{3}y-\frac{2}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\left(-7\right)-\frac{2}{3}$$

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

$$x=\frac{-7-2}{3}$$

Tāpiri -\frac{2}{3} ki te -\frac{7}{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.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-7$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-1\\5&-2\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=-7$$

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.

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

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.

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

Whakarūnātia.

$$15x-5y+10=0,15x-6y+3=0$$

Me tango 15x-6y+3=0 mai i 15x-5y+10=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-5y+6y+10-3=0$$

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.

$$-5y+6y+10-3=0$$

Tāpiri -5y ki te 6y.

$$y+10-3=0$$

Tāpiri 10 ki te -3.

$$y+7=0$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$y=-7$$

Whakaurua te -7 mō y ki 5x-2y+1=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-2\left(-7\right)+1=0$$

Whakareatia -2 ki te -7.

$$5x+14+1=0$$

Tāpiri 14 ki te 1.

$$5x+15=0$$

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

$$5x=-15$$

Whakawehea ngā taha e rua ki te 5.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-7$$