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

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

Whakaoti mō x, y

$$x=\frac{8}{15}\approx 0.533333333$$

$$y=\frac{13}{15}\approx 0.866666667$$

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.

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

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

$$4x+y=3$$

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

$$4x=-y+3$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

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

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

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

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

Tāpiri -\frac{3y}{4} ki te -3y.

$$-\frac{15}{4}y+\frac{9}{4}=-1$$

Me tango \frac{9}{4} mai i ngā taha e rua o te whārite.

$$-\frac{15}{4}y=-\frac{13}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{15}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{13}{15}$$

Whakaurua te \frac{13}{15} mō y ki x=-\frac{1}{4}y+\frac{3}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{4}\times \frac{13}{15}+\frac{3}{4}$$

Whakareatia -\frac{1}{4} ki te \frac{13}{15} 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.

$$x=-\frac{13}{60}+\frac{3}{4}$$

Tāpiri \frac{3}{4} ki te -\frac{13}{60} 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.

$$x=\frac{8}{15}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{15},y=\frac{13}{15}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{1}{15}\\\frac{1}{5}&-\frac{4}{15}\end{matrix}\right)\left(\begin{matrix}3\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 3+\frac{1}{15}\left(-1\right)\\\frac{1}{5}\times 3-\frac{4}{15}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{15}\\\frac{13}{15}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{8}{15},y=\frac{13}{15}$$

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.

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

Kia ōrite ai a 4x 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 4.

$$3\times 4x+3y=3\times 3,4\times 3x+4\left(-3\right)y=4\left(-1\right)$$

Whakarūnātia.

$$12x+3y=9,12x-12y=-4$$

Me tango 12x-12y=-4 mai i 12x+3y=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+3y+12y=9+4$$

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.

$$3y+12y=9+4$$

Tāpiri 3y ki te 12y.

$$15y=9+4$$

Tāpiri 9 ki te 4.

$$15y=13$$

Whakawehea ngā taha e rua ki te 15.

$$y=\frac{13}{15}$$

Whakaurua te \frac{13}{15} mō y ki 3x-3y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-3\times \frac{13}{15}=-1$$

Whakareatia -3 ki te \frac{13}{15}.

$$3x-\frac{13}{5}=-1$$

Me tāpiri \frac{13}{5} ki ngā taha e rua o te whārite.

$$3x=\frac{8}{5}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{8}{15}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{15},y=\frac{13}{15}$$