Whakaoti i te {l}{4x+2y=25,2}{5y+x=32}

$$\left\{ \begin{array} { l } { 4 x + 2 y = 25,2 } \\ { 5 y + x = 32 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{31}{9} = 3\frac{4}{9} \approx 3.444444444$$

$$y = \frac{257}{45} = 5\frac{32}{45} \approx 5.711111111$$

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.

$$4x+2y=25.2,x+5y=32$$

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

$$4x+2y=25.2$$

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

$$4x=-2y+25.2$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-2y+25.2\right)$$

Whakareatia \frac{1}{4} ki te -2y+25.2.

$$x=-\frac{1}{2}y+\frac{63}{10}$$

Whakakapia te -\frac{y}{2}+\frac{63}{10} mō te x ki tērā atu whārite, x+5y=32.

$$-\frac{1}{2}y+\frac{63}{10}+5y=32$$

Tāpiri -\frac{y}{2} ki te 5y.

$$\frac{9}{2}y+\frac{63}{10}=32$$

Me tango \frac{63}{10} mai i ngā taha e rua o te whārite.

$$\frac{9}{2}y=\frac{257}{10}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{257}{45}$$

Whakaurua te \frac{257}{45} mō y ki x=-\frac{1}{2}y+\frac{63}{10}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times \frac{257}{45}+\frac{63}{10}$$

Whakareatia -\frac{1}{2} ki te \frac{257}{45} 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.

$$x=-\frac{257}{90}+\frac{63}{10}$$

Tāpiri \frac{63}{10} ki te -\frac{257}{90} 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.

$$x=\frac{31}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{31}{9},y=\frac{257}{45}$$

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.

$$4x+2y=25.2,x+5y=32$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&2\\1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25.2\\32\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&2\\1&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&2\\1&5\end{matrix}\right))\left(\begin{matrix}4&2\\1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\1&5\end{matrix}\right))\left(\begin{matrix}25.2\\32\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\1&5\end{matrix}\right))\left(\begin{matrix}25.2\\32\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\1&5\end{matrix}\right))\left(\begin{matrix}25.2\\32\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-2}&-\frac{2}{4\times 5-2}\\-\frac{1}{4\times 5-2}&\frac{4}{4\times 5-2}\end{matrix}\right)\left(\begin{matrix}25.2\\32\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{18}&-\frac{1}{9}\\-\frac{1}{18}&\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}25.2\\32\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{18}\times 25.2-\frac{1}{9}\times 32\\-\frac{1}{18}\times 25.2+\frac{2}{9}\times 32\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{31}{9}\\\frac{257}{45}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{31}{9},y=\frac{257}{45}$$

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.

$$4x+2y=25.2,x+5y=32$$

Kia ōrite ai a 4x 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 4.

$$4x+2y=25.2,4x+4\times 5y=4\times 32$$

Whakarūnātia.

$$4x+2y=25.2,4x+20y=128$$

Me tango 4x+20y=128 mai i 4x+2y=25.2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+2y-20y=25.2-128$$

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.

$$2y-20y=25.2-128$$

Tāpiri 2y ki te -20y.

$$-18y=25.2-128$$

Tāpiri 25.2 ki te -128.

$$-18y=-102.8$$

Whakawehea ngā taha e rua ki te -18.

$$y=\frac{257}{45}$$

Whakaurua te \frac{257}{45} mō y ki x+5y=32. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+5\times \frac{257}{45}=32$$

Whakareatia 5 ki te \frac{257}{45}.

$$x+\frac{257}{9}=32$$

Me tango \frac{257}{9} mai i ngā taha e rua o te whārite.

$$x=\frac{31}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{31}{9},y=\frac{257}{45}$$