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

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

Whakaoti mō x, y

$$x=-\frac{3}{5}=-0.6$$

$$y = \frac{12}{5} = 2\frac{2}{5} = 2.4$$

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.

$$3x+2y=3,4x+y=0$$

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

$$3x+2y=3$$

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

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

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te -\frac{2y}{3}+1 mō te x ki tērā atu whārite, 4x+y=0.

$$4\left(-\frac{2}{3}y+1\right)+y=0$$

Whakareatia 4 ki te -\frac{2y}{3}+1.

$$-\frac{8}{3}y+4+y=0$$

Tāpiri -\frac{8y}{3} ki te y.

$$-\frac{5}{3}y+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-\frac{5}{3}y=-4$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{12}{5}$$

Whakaurua te \frac{12}{5} mō y ki x=-\frac{2}{3}y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{12}{5}+1$$

Whakareatia -\frac{2}{3} ki te \frac{12}{5} 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.

$$x=-\frac{8}{5}+1$$

Tāpiri 1 ki te -\frac{8}{5}.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{3}{5},y=\frac{12}{5}$$

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.

$$3x+2y=3,4x+y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&2\\4&1\end{matrix}\right).

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-2\times 4}&-\frac{2}{3-2\times 4}\\-\frac{4}{3-2\times 4}&\frac{3}{3-2\times 4}\end{matrix}\right)\left(\begin{matrix}3\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 3\\\frac{4}{5}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5}\\\frac{12}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{3}{5},y=\frac{12}{5}$$

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.

$$3x+2y=3,4x+y=0$$

Kia ōrite ai a 3x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

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

Whakarūnātia.

$$12x+8y=12,12x+3y=0$$

Me tango 12x+3y=0 mai i 12x+8y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+8y-3y=12$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-3y=12$$

Tāpiri 8y ki te -3y.

$$5y=12$$

Whakawehea ngā taha e rua ki te 5.

$$y=\frac{12}{5}$$

Whakaurua te \frac{12}{5} mō y ki 4x+y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+\frac{12}{5}=0$$

Me tango \frac{12}{5} mai i ngā taha e rua o te whārite.

$$4x=-\frac{12}{5}$$

Whakawehea ngā taha e rua ki te 4.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{3}{5},y=\frac{12}{5}$$