Whakaoti i te {l}{3x+2y=12}{4x-y=11}

$$\left. \begin{array} { l } { 3 x + 2 y = 12 } \\ { 4 x - y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{34}{11} = 3\frac{1}{11} \approx 3.090909091$$

$$y = \frac{15}{11} = 1\frac{4}{11} \approx 1.363636364$$

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.

$$3x+2y=12,4x-y=11$$

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

$$3x+2y=12$$

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

$$3x=-2y+12$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y+12\right)$$

Whakareatia \frac{1}{3} ki te -2y+12.

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

Whakakapia te -\frac{2y}{3}+4 mō te x ki tērā atu whārite, 4x-y=11.

$$4\left(-\frac{2}{3}y+4\right)-y=11$$

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

$$-\frac{8}{3}y+16-y=11$$

Tāpiri -\frac{8y}{3} ki te -y.

$$-\frac{11}{3}y+16=11$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$-\frac{11}{3}y=-5$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{15}{11}$$

Whakaurua te \frac{15}{11} mō y ki x=-\frac{2}{3}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{15}{11}+4$$

Whakareatia -\frac{2}{3} ki te \frac{15}{11} 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.

$$x=-\frac{10}{11}+4$$

Tāpiri 4 ki te -\frac{10}{11}.

$$x=\frac{34}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{34}{11},y=\frac{15}{11}$$

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.

$$3x+2y=12,4x-y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\11\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&2\\4&-1\end{matrix}\right))\left(\begin{matrix}3&2\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\4&-1\end{matrix}\right))\left(\begin{matrix}12\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\4&-1\end{matrix}\right))\left(\begin{matrix}12\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\4&-1\end{matrix}\right))\left(\begin{matrix}12\\11\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-2\times 4}&-\frac{2}{3\left(-1\right)-2\times 4}\\-\frac{4}{3\left(-1\right)-2\times 4}&\frac{3}{3\left(-1\right)-2\times 4}\end{matrix}\right)\left(\begin{matrix}12\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{2}{11}\\\frac{4}{11}&-\frac{3}{11}\end{matrix}\right)\left(\begin{matrix}12\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 12+\frac{2}{11}\times 11\\\frac{4}{11}\times 12-\frac{3}{11}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{34}{11}\\\frac{15}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{34}{11},y=\frac{15}{11}$$

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.

$$3x+2y=12,4x-y=11$$

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.

$$4\times 3x+4\times 2y=4\times 12,3\times 4x+3\left(-1\right)y=3\times 11$$

Whakarūnātia.

$$12x+8y=48,12x-3y=33$$

Me tango 12x-3y=33 mai i 12x+8y=48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+8y+3y=48-33$$

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.

$$8y+3y=48-33$$

Tāpiri 8y ki te 3y.

$$11y=48-33$$

Tāpiri 48 ki te -33.

$$11y=15$$

Whakawehea ngā taha e rua ki te 11.

$$y=\frac{15}{11}$$

Whakaurua te \frac{15}{11} mō y ki 4x-y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-\frac{15}{11}=11$$

Me tāpiri \frac{15}{11} ki ngā taha e rua o te whārite.

$$4x=\frac{136}{11}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{34}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{34}{11},y=\frac{15}{11}$$