Whakaoti i te {l}{y+3x=56}{x+4y=34}

$$\left. \begin{array} { l } { y + 3 x = 56 } \\ { x + 4 y = 34 } \end{array} \right.$$

Whakaoti mō y, x

$$x = \frac{190}{11} = 17\frac{3}{11} \approx 17.272727273$$

$$y = \frac{46}{11} = 4\frac{2}{11} \approx 4.181818182$$

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.

$$y+3x=56,4y+x=34$$

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

$$y+3x=56$$

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

$$y=-3x+56$$

Whakakapia te -3x+56 mō te y ki tērā atu whārite, 4y+x=34.

$$4\left(-3x+56\right)+x=34$$

Whakareatia 4 ki te -3x+56.

$$-12x+224+x=34$$

Tāpiri -12x ki te x.

$$-11x+224=34$$

Me tango 224 mai i ngā taha e rua o te whārite.

$$-11x=-190$$

Whakawehea ngā taha e rua ki te -11.

$$x=\frac{190}{11}$$

Whakaurua te \frac{190}{11} mō x ki y=-3x+56. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-3\times \frac{190}{11}+56$$

Whakareatia -3 ki te \frac{190}{11}.

$$y=-\frac{570}{11}+56$$

Tāpiri 56 ki te -\frac{570}{11}.

$$y=\frac{46}{11}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{46}{11},x=\frac{190}{11}$$

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.

$$y+3x=56,4y+x=34$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\4&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}56\\34\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&3\\4&1\end{matrix}\right))\left(\begin{matrix}1&3\\4&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\4&1\end{matrix}\right))\left(\begin{matrix}56\\34\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\4&1\end{matrix}\right))\left(\begin{matrix}56\\34\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\4&1\end{matrix}\right))\left(\begin{matrix}56\\34\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-3\times 4}&-\frac{3}{1-3\times 4}\\-\frac{4}{1-3\times 4}&\frac{1}{1-3\times 4}\end{matrix}\right)\left(\begin{matrix}56\\34\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{11}&\frac{3}{11}\\\frac{4}{11}&-\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}56\\34\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{11}\times 56+\frac{3}{11}\times 34\\\frac{4}{11}\times 56-\frac{1}{11}\times 34\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{46}{11}\\\frac{190}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{46}{11},x=\frac{190}{11}$$

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.

$$y+3x=56,4y+x=34$$

Kia ōrite ai a y me 4y, 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 1.

$$4y+4\times 3x=4\times 56,4y+x=34$$

Whakarūnātia.

$$4y+12x=224,4y+x=34$$

Me tango 4y+x=34 mai i 4y+12x=224 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4y-4y+12x-x=224-34$$

Tāpiri 4y ki te -4y. Ka whakakore atu ngā kupu 4y me -4y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12x-x=224-34$$

Tāpiri 12x ki te -x.

$$11x=224-34$$

Tāpiri 224 ki te -34.

$$11x=190$$

Whakawehea ngā taha e rua ki te 11.

$$x=\frac{190}{11}$$

Whakaurua te \frac{190}{11} mō x ki 4y+x=34. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$4y+\frac{190}{11}=34$$

Me tango \frac{190}{11} mai i ngā taha e rua o te whārite.

$$4y=\frac{184}{11}$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{46}{11}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{46}{11},x=\frac{190}{11}$$