Whakaoti i te 3x+10y=102,3x+1y=84

$$3 x + 10 y = 102,3 x + 1 y = 84$$

Whakaoti mō x, y

$$x = \frac{82}{3} = 27\frac{1}{3} \approx 27.333333333$$

$$y=2$$

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.

$$3x+10y=102,3x+y=84$$

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

$$3x+10y=102$$

Me tango 10y mai i ngā taha e rua o te whārite.

$$3x=-10y+102$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-10y+102\right)$$

Whakareatia \frac{1}{3} ki te -10y+102.

$$x=-\frac{10}{3}y+34$$

Whakakapia te -\frac{10y}{3}+34 mō te x ki tērā atu whārite, 3x+y=84.

$$3\left(-\frac{10}{3}y+34\right)+y=84$$

Whakareatia 3 ki te -\frac{10y}{3}+34.

$$-10y+102+y=84$$

Tāpiri -10y ki te y.

$$-9y+102=84$$

Me tango 102 mai i ngā taha e rua o te whārite.

$$-9y=-18$$

Whakawehea ngā taha e rua ki te -9.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{10}{3}y+34. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{10}{3}\times 2+34$$

Whakareatia -\frac{10}{3} ki te 2.

$$x=-\frac{20}{3}+34$$

Tāpiri 34 ki te -\frac{20}{3}.

$$x=\frac{82}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{82}{3},y=2$$

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.

$$3x+10y=102,3x+y=84$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&10\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}102\\84\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&10\\3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&10\\3&1\end{matrix}\right))\left(\begin{matrix}3&10\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&10\\3&1\end{matrix}\right))\left(\begin{matrix}102\\84\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&10\\3&1\end{matrix}\right))\left(\begin{matrix}102\\84\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&10\\3&1\end{matrix}\right))\left(\begin{matrix}102\\84\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-10\times 3}&-\frac{10}{3-10\times 3}\\-\frac{3}{3-10\times 3}&\frac{3}{3-10\times 3}\end{matrix}\right)\left(\begin{matrix}102\\84\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{27}&\frac{10}{27}\\\frac{1}{9}&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}102\\84\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{27}\times 102+\frac{10}{27}\times 84\\\frac{1}{9}\times 102-\frac{1}{9}\times 84\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{82}{3}\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{82}{3},y=2$$

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.

$$3x+10y=102,3x+y=84$$

Me tango 3x+y=84 mai i 3x+10y=102 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+10y-y=102-84$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y-y=102-84$$

Tāpiri 10y ki te -y.

$$9y=102-84$$

Tāpiri 102 ki te -84.

$$9y=18$$

Whakawehea ngā taha e rua ki te 9.

$$y=2$$

Whakaurua te 2 mō y ki 3x+y=84. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2=84$$

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

$$3x=82$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{82}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{82}{3},y=2$$