Whakaoti i te {c}{2x+5y=8}{x-3y=3}

$$\left\{ \begin{array} { c } { 2 x + 5 y = 8 } \\ { x - 3 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{39}{11} = 3\frac{6}{11} \approx 3.545454545$$

$$y=\frac{2}{11}\approx 0.181818182$$

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.

$$2x+5y=8,x-3y=3$$

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

$$2x+5y=8$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$2x=-5y+8$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y+8\right)$$

Whakareatia \frac{1}{2} ki te -5y+8.

$$x=-\frac{5}{2}y+4$$

Whakakapia te -\frac{5y}{2}+4 mō te x ki tērā atu whārite, x-3y=3.

$$-\frac{5}{2}y+4-3y=3$$

Tāpiri -\frac{5y}{2} ki te -3y.

$$-\frac{11}{2}y+4=3$$

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

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

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{2}{11}$$

Whakaurua te \frac{2}{11} mō y ki x=-\frac{5}{2}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}\times \frac{2}{11}+4$$

Whakareatia -\frac{5}{2} ki te \frac{2}{11} 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.

$$x=-\frac{5}{11}+4$$

Tāpiri 4 ki te -\frac{5}{11}.

$$x=\frac{39}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{39}{11},y=\frac{2}{11}$$

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.

$$2x+5y=8,x-3y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&5\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&5\\1&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\1&-3\end{matrix}\right))\left(\begin{matrix}2&5\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\1&-3\end{matrix}\right))\left(\begin{matrix}8\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\1&-3\end{matrix}\right))\left(\begin{matrix}8\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\1&-3\end{matrix}\right))\left(\begin{matrix}8\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2\left(-3\right)-5}&-\frac{5}{2\left(-3\right)-5}\\-\frac{1}{2\left(-3\right)-5}&\frac{2}{2\left(-3\right)-5}\end{matrix}\right)\left(\begin{matrix}8\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}&\frac{5}{11}\\\frac{1}{11}&-\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}8\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}\times 8+\frac{5}{11}\times 3\\\frac{1}{11}\times 8-\frac{2}{11}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{39}{11}\\\frac{2}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{39}{11},y=\frac{2}{11}$$

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.

$$2x+5y=8,x-3y=3$$

Kia ōrite ai a 2x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$2x+5y=8,2x+2\left(-3\right)y=2\times 3$$

Whakarūnātia.

$$2x+5y=8,2x-6y=6$$

Me tango 2x-6y=6 mai i 2x+5y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+5y+6y=8-6$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y+6y=8-6$$

Tāpiri 5y ki te 6y.

$$11y=8-6$$

Tāpiri 8 ki te -6.

$$11y=2$$

Whakawehea ngā taha e rua ki te 11.

$$y=\frac{2}{11}$$

Whakaurua te \frac{2}{11} mō y ki x-3y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\times \frac{2}{11}=3$$

Whakareatia -3 ki te \frac{2}{11}.

$$x-\frac{6}{11}=3$$

Me tāpiri \frac{6}{11} ki ngā taha e rua o te whārite.

$$x=\frac{39}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{39}{11},y=\frac{2}{11}$$