Whakaoti i te {l}{1x+4y=280}{4x+1y=124}

$$\left. \begin{array} { l } { 1 x + 4 y = 280 } \\ { 4 x + 1 y = 124 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{72}{5} = 14\frac{2}{5} = 14.4$$

$$y = \frac{332}{5} = 66\frac{2}{5} = 66.4$$

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.

$$x+4y=280,4x+y=124$$

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

$$x+4y=280$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$x=-4y+280$$

Whakakapia te -4y+280 mō te x ki tērā atu whārite, 4x+y=124.

$$4\left(-4y+280\right)+y=124$$

Whakareatia 4 ki te -4y+280.

$$-16y+1120+y=124$$

Tāpiri -16y ki te y.

$$-15y+1120=124$$

Me tango 1120 mai i ngā taha e rua o te whārite.

$$-15y=-996$$

Whakawehea ngā taha e rua ki te -15.

$$y=\frac{332}{5}$$

Whakaurua te \frac{332}{5} mō y ki x=-4y+280. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4\times \frac{332}{5}+280$$

Whakareatia -4 ki te \frac{332}{5}.

$$x=-\frac{1328}{5}+280$$

Tāpiri 280 ki te -\frac{1328}{5}.

$$x=\frac{72}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{72}{5},y=\frac{332}{5}$$

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.

$$x+4y=280,4x+y=124$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&4\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}280\\124\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&4\\4&1\end{matrix}\right))\left(\begin{matrix}1&4\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\4&1\end{matrix}\right))\left(\begin{matrix}280\\124\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\4&1\end{matrix}\right))\left(\begin{matrix}280\\124\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\4&1\end{matrix}\right))\left(\begin{matrix}280\\124\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-4\times 4}&-\frac{4}{1-4\times 4}\\-\frac{4}{1-4\times 4}&\frac{1}{1-4\times 4}\end{matrix}\right)\left(\begin{matrix}280\\124\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{15}&\frac{4}{15}\\\frac{4}{15}&-\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}280\\124\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{15}\times 280+\frac{4}{15}\times 124\\\frac{4}{15}\times 280-\frac{1}{15}\times 124\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{72}{5}\\\frac{332}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{72}{5},y=\frac{332}{5}$$

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.

$$x+4y=280,4x+y=124$$

Kia ōrite ai a x 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 1.

$$4x+4\times 4y=4\times 280,4x+y=124$$

Whakarūnātia.

$$4x+16y=1120,4x+y=124$$

Me tango 4x+y=124 mai i 4x+16y=1120 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+16y-y=1120-124$$

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

$$16y-y=1120-124$$

Tāpiri 16y ki te -y.

$$15y=1120-124$$

Tāpiri 1120 ki te -124.

$$15y=996$$

Whakawehea ngā taha e rua ki te 15.

$$y=\frac{332}{5}$$

Whakaurua te \frac{332}{5} mō y ki 4x+y=124. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+\frac{332}{5}=124$$

Me tango \frac{332}{5} mai i ngā taha e rua o te whārite.

$$4x=\frac{288}{5}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{72}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{72}{5},y=\frac{332}{5}$$