Whakaoti i te {l}{4x+3y+14=0}{2x+5y+16=0}

$$\left\{ \begin{array} { l } { 4 x + 3 y + 14 = 0 } \\ { 2 x + 5 y + 16 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{11}{7} = -1\frac{4}{7} \approx -1.571428571$$

$$y = -\frac{18}{7} = -2\frac{4}{7} \approx -2.571428571$$

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.

$$4x+3y+14=0,2x+5y+16=0$$

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

$$4x+3y+14=0$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$4x+3y=-14$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$4x=-3y-14$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-3y-14\right)$$

Whakareatia \frac{1}{4} ki te -3y-14.

$$x=-\frac{3}{4}y-\frac{7}{2}$$

Whakakapia te -\frac{3y}{4}-\frac{7}{2} mō te x ki tērā atu whārite, 2x+5y+16=0.

$$2\left(-\frac{3}{4}y-\frac{7}{2}\right)+5y+16=0$$

Whakareatia 2 ki te -\frac{3y}{4}-\frac{7}{2}.

$$-\frac{3}{2}y-7+5y+16=0$$

Tāpiri -\frac{3y}{2} ki te 5y.

$$\frac{7}{2}y-7+16=0$$

Tāpiri -7 ki te 16.

$$\frac{7}{2}y+9=0$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$\frac{7}{2}y=-9$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{18}{7}$$

Whakaurua te -\frac{18}{7} mō y ki x=-\frac{3}{4}y-\frac{7}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\left(-\frac{18}{7}\right)-\frac{7}{2}$$

Whakareatia -\frac{3}{4} ki te -\frac{18}{7} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{27}{14}-\frac{7}{2}$$

Tāpiri -\frac{7}{2} ki te \frac{27}{14} 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.

$$x=-\frac{11}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{11}{7},y=-\frac{18}{7}$$

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.

$$4x+3y+14=0,2x+5y+16=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&3\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-14\\-16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&3\\2&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&3\\2&5\end{matrix}\right))\left(\begin{matrix}4&3\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\2&5\end{matrix}\right))\left(\begin{matrix}-14\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\2&5\end{matrix}\right))\left(\begin{matrix}-14\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\2&5\end{matrix}\right))\left(\begin{matrix}-14\\-16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-3\times 2}&-\frac{3}{4\times 5-3\times 2}\\-\frac{2}{4\times 5-3\times 2}&\frac{4}{4\times 5-3\times 2}\end{matrix}\right)\left(\begin{matrix}-14\\-16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{14}&-\frac{3}{14}\\-\frac{1}{7}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}-14\\-16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{14}\left(-14\right)-\frac{3}{14}\left(-16\right)\\-\frac{1}{7}\left(-14\right)+\frac{2}{7}\left(-16\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{7}\\-\frac{18}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{11}{7},y=-\frac{18}{7}$$

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.

$$4x+3y+14=0,2x+5y+16=0$$

Kia ōrite ai a 4x 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 4.

$$2\times 4x+2\times 3y+2\times 14=0,4\times 2x+4\times 5y+4\times 16=0$$

Whakarūnātia.

$$8x+6y+28=0,8x+20y+64=0$$

Me tango 8x+20y+64=0 mai i 8x+6y+28=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+6y-20y+28-64=0$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-20y+28-64=0$$

Tāpiri 6y ki te -20y.

$$-14y+28-64=0$$

Tāpiri 28 ki te -64.

$$-14y-36=0$$

Me tāpiri 36 ki ngā taha e rua o te whārite.

$$-14y=36$$

Whakawehea ngā taha e rua ki te -14.

$$y=-\frac{18}{7}$$

Whakaurua te -\frac{18}{7} mō y ki 2x+5y+16=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+5\left(-\frac{18}{7}\right)+16=0$$

Whakareatia 5 ki te -\frac{18}{7}.

$$2x-\frac{90}{7}+16=0$$

Tāpiri -\frac{90}{7} ki te 16.

$$2x+\frac{22}{7}=0$$

Me tango \frac{22}{7} mai i ngā taha e rua o te whārite.

$$2x=-\frac{22}{7}$$

Whakawehea ngā taha e rua ki te 2.

$$x=-\frac{11}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{11}{7},y=-\frac{18}{7}$$