Whakaoti i te {l}{4x+3y=0}{3y+3x=1}

$$\left. \begin{array} { l } { 4 x + 3 y = 0 } \\ { 3 y + 3 x = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

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.

$$4x+3y=0,3x+3y=1$$

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

$$4x+3y=0$$

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

$$4x=-3y$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

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

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

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

$$-\frac{9}{4}y+3y=1$$

Tāpiri -\frac{9y}{4} ki te 3y.

$$\frac{3}{4}y=1$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

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

$$x=-\frac{3}{4}\times \frac{4}{3}$$

Whakareatia -\frac{3}{4} ki te \frac{4}{3} 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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=\frac{4}{3}$$

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.

$$4x+3y=0,3x+3y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=\frac{4}{3}$$

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.

$$4x+3y=0,3x+3y=1$$

Me tango 3x+3y=1 mai i 4x+3y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-3x+3y-3y=-1$$

Tāpiri 3y ki te -3y. Ka whakakore atu ngā kupu 3y me -3y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4x-3x=-1$$

Tāpiri 4x ki te -3x.

$$x=-1$$

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

$$3\left(-1\right)+3y=1$$

Whakareatia 3 ki te -1.

$$-3+3y=1$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$3y=4$$

Whakawehea ngā taha e rua ki te 3.

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

Kua oti te pūnaha te whakatau.

$$x=-1,y=\frac{4}{3}$$