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

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

Whakaoti mō x, y

$$x=\frac{1}{8}=0.125$$

$$y=\frac{5}{8}=0.625$$

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.

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

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

$$3x+y=1$$

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

$$3x=-y+1$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

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

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

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

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

Tāpiri -\frac{4y}{3} ki te 4y.

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

Me tango \frac{4}{3} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{5}{8}$$

Whakaurua te \frac{5}{8} mō y ki x=-\frac{1}{3}y+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times \frac{5}{8}+\frac{1}{3}$$

Whakareatia -\frac{1}{3} ki te \frac{5}{8} 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.

$$x=-\frac{5}{24}+\frac{1}{3}$$

Tāpiri \frac{1}{3} ki te -\frac{5}{24} 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.

$$x=\frac{1}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{8},y=\frac{5}{8}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{8}\\-\frac{1}{2}&\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}1\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}-\frac{1}{8}\times 3\\-\frac{1}{2}+\frac{3}{8}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}\\\frac{5}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{8},y=\frac{5}{8}$$

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.

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

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.

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

Whakarūnātia.

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

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

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

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.

$$4y-12y=4-9$$

Tāpiri 4y ki te -12y.

$$-8y=4-9$$

Tāpiri 4 ki te -9.

$$-8y=-5$$

Whakawehea ngā taha e rua ki te -8.

$$y=\frac{5}{8}$$

Whakaurua te \frac{5}{8} mō y ki 4x+4y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+4\times \frac{5}{8}=3$$

Whakareatia 4 ki te \frac{5}{8}.

$$4x+\frac{5}{2}=3$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$4x=\frac{1}{2}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{8},y=\frac{5}{8}$$