Whakaoti i te {l}{x+y=1700}{15x+20y=65}

$$\left. \begin{array} { l } { x + y = 1700 } \\ { 15 x + 20 y = 65 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6787$$

$$y=-5087$$

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.

$$x+y=1700,15x+20y=65$$

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

$$x+y=1700$$

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

$$x=-y+1700$$

Whakakapia te -y+1700 mō te x ki tērā atu whārite, 15x+20y=65.

$$15\left(-y+1700\right)+20y=65$$

Whakareatia 15 ki te -y+1700.

$$-15y+25500+20y=65$$

Tāpiri -15y ki te 20y.

$$5y+25500=65$$

Me tango 25500 mai i ngā taha e rua o te whārite.

$$5y=-25435$$

Whakawehea ngā taha e rua ki te 5.

$$y=-5087$$

Whakaurua te -5087 mō y ki x=-y+1700. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\left(-5087\right)+1700$$

Whakareatia -1 ki te -5087.

$$x=5087+1700$$

Tāpiri 1700 ki te 5087.

$$x=6787$$

Kua oti te pūnaha te whakatau.

$$x=6787,y=-5087$$

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.

$$x+y=1700,15x+20y=65$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\15&20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1700\\65\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\15&20\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\15&20\end{matrix}\right))\left(\begin{matrix}1&1\\15&20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\15&20\end{matrix}\right))\left(\begin{matrix}1700\\65\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\15&20\end{matrix}\right))\left(\begin{matrix}1700\\65\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\15&20\end{matrix}\right))\left(\begin{matrix}1700\\65\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{20-15}&-\frac{1}{20-15}\\-\frac{15}{20-15}&\frac{1}{20-15}\end{matrix}\right)\left(\begin{matrix}1700\\65\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4&-\frac{1}{5}\\-3&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}1700\\65\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\times 1700-\frac{1}{5}\times 65\\-3\times 1700+\frac{1}{5}\times 65\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6787\\-5087\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6787,y=-5087$$

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.

$$x+y=1700,15x+20y=65$$

Kia ōrite ai a x me 15x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 15 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$15x+15y=15\times 1700,15x+20y=65$$

Whakarūnātia.

$$15x+15y=25500,15x+20y=65$$

Me tango 15x+20y=65 mai i 15x+15y=25500 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+15y-20y=25500-65$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y-20y=25500-65$$

Tāpiri 15y ki te -20y.

$$-5y=25500-65$$

Tāpiri 25500 ki te -65.

$$-5y=25435$$

Whakawehea ngā taha e rua ki te -5.

$$y=-5087$$

Whakaurua te -5087 mō y ki 15x+20y=65. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$15x+20\left(-5087\right)=65$$

Whakareatia 20 ki te -5087.

$$15x-101740=65$$

Me tāpiri 101740 ki ngā taha e rua o te whārite.

$$15x=101805$$

Whakawehea ngā taha e rua ki te 15.

$$x=6787$$

Kua oti te pūnaha te whakatau.

$$x=6787,y=-5087$$