Whakaoti i te {l}{2(3x+y)=1/2}{1/2(2x+8y)=3/2}

$$\left. \begin{array} { l } { 2 ( 3 x + y ) = \frac { 1 } { 2 } } \\ { \frac { 1 } { 2 } ( 2 x + 8 y ) = \frac { 3 } { 2 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{1}{22}\approx -0.045454545$$

$$y=\frac{17}{44}\approx 0.386363636$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawehea ngā taha e rua ki te 2.

$$3x+y=\frac{\frac{1}{2}}{2}$$

Tuhia te \frac{\frac{1}{2}}{2} hei hautanga kotahi.

$$3x+y=\frac{1}{2\times 2}$$

Whakareatia te 2 ki te 2, ka 4.

$$3x+y=\frac{1}{4}$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua ki te 2, te tau utu o \frac{1}{2}.

$$2x+8y=\frac{3}{2}\times 2$$

Whakareatia te \frac{3}{2} ki te 2, ka 3.

$$2x+8y=3$$

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.

$$3x+y=\frac{1}{4},2x+8y=3$$

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

$$3x+y=\frac{1}{4}$$

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

$$3x=-y+\frac{1}{4}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-y+\frac{1}{4}\right)$$

Whakareatia \frac{1}{3} ki te -y+\frac{1}{4}.

$$x=-\frac{1}{3}y+\frac{1}{12}$$

Whakakapia te -\frac{y}{3}+\frac{1}{12} mō te x ki tērā atu whārite, 2x+8y=3.

$$2\left(-\frac{1}{3}y+\frac{1}{12}\right)+8y=3$$

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

$$-\frac{2}{3}y+\frac{1}{6}+8y=3$$

Tāpiri -\frac{2y}{3} ki te 8y.

$$\frac{22}{3}y+\frac{1}{6}=3$$

Me tango \frac{1}{6} mai i ngā taha e rua o te whārite.

$$\frac{22}{3}y=\frac{17}{6}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{22}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{17}{44}$$

Whakaurua te \frac{17}{44} mō y ki x=-\frac{1}{3}y+\frac{1}{12}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times \frac{17}{44}+\frac{1}{12}$$

Whakareatia -\frac{1}{3} ki te \frac{17}{44} 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.

$$x=-\frac{17}{132}+\frac{1}{12}$$

Tāpiri \frac{1}{12} ki te -\frac{17}{132} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{1}{22}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{22},y=\frac{17}{44}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawehea ngā taha e rua ki te 2.

$$3x+y=\frac{\frac{1}{2}}{2}$$

Tuhia te \frac{\frac{1}{2}}{2} hei hautanga kotahi.

$$3x+y=\frac{1}{2\times 2}$$

Whakareatia te 2 ki te 2, ka 4.

$$3x+y=\frac{1}{4}$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua ki te 2, te tau utu o \frac{1}{2}.

$$2x+8y=\frac{3}{2}\times 2$$

Whakareatia te \frac{3}{2} ki te 2, ka 3.

$$2x+8y=3$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3x+y=\frac{1}{4},2x+8y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\2&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\2&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&1\\2&8\end{matrix}\right))\left(\begin{matrix}3&1\\2&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&8\end{matrix}\right))\left(\begin{matrix}\frac{1}{4}\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&8\end{matrix}\right))\left(\begin{matrix}\frac{1}{4}\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&8\end{matrix}\right))\left(\begin{matrix}\frac{1}{4}\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3\times 8-2}&-\frac{1}{3\times 8-2}\\-\frac{2}{3\times 8-2}&\frac{3}{3\times 8-2}\end{matrix}\right)\left(\begin{matrix}\frac{1}{4}\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}&-\frac{1}{22}\\-\frac{1}{11}&\frac{3}{22}\end{matrix}\right)\left(\begin{matrix}\frac{1}{4}\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}\times \frac{1}{4}-\frac{1}{22}\times 3\\-\frac{1}{11}\times \frac{1}{4}+\frac{3}{22}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{22}\\\frac{17}{44}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1}{22},y=\frac{17}{44}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawehea ngā taha e rua ki te 2.

$$3x+y=\frac{\frac{1}{2}}{2}$$

Tuhia te \frac{\frac{1}{2}}{2} hei hautanga kotahi.

$$3x+y=\frac{1}{2\times 2}$$

Whakareatia te 2 ki te 2, ka 4.

$$3x+y=\frac{1}{4}$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua ki te 2, te tau utu o \frac{1}{2}.

$$2x+8y=\frac{3}{2}\times 2$$

Whakareatia te \frac{3}{2} ki te 2, ka 3.

$$2x+8y=3$$

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.

$$3x+y=\frac{1}{4},2x+8y=3$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2y=2\times \frac{1}{4},3\times 2x+3\times 8y=3\times 3$$

Whakarūnātia.

$$6x+2y=\frac{1}{2},6x+24y=9$$

Me tango 6x+24y=9 mai i 6x+2y=\frac{1}{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+2y-24y=\frac{1}{2}-9$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y-24y=\frac{1}{2}-9$$

Tāpiri 2y ki te -24y.

$$-22y=\frac{1}{2}-9$$

Tāpiri \frac{1}{2} ki te -9.

$$-22y=-\frac{17}{2}$$

Whakawehea ngā taha e rua ki te -22.

$$y=\frac{17}{44}$$

Whakaurua te \frac{17}{44} mō y ki 2x+8y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+8\times \frac{17}{44}=3$$

Whakareatia 8 ki te \frac{17}{44}.

$$2x+\frac{34}{11}=3$$

Me tango \frac{34}{11} mai i ngā taha e rua o te whārite.

$$2x=-\frac{1}{11}$$

Whakawehea ngā taha e rua ki te 2.

$$x=-\frac{1}{22}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{22},y=\frac{17}{44}$$