Whakaoti i te {l}{25x+y=9}{1.6x+0.2y=13}

$$\left. \begin{array} { l } { 25 x + y = 9 } \\ { 1.6 x + 0.2 y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{56}{17} = -3\frac{5}{17} \approx -3.294117647$$

$$y = \frac{1553}{17} = 91\frac{6}{17} \approx 91.352941176$$

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.

$$25x+y=9,1.6x+0.2y=13$$

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

$$25x+y=9$$

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

$$25x=-y+9$$

Whakawehea ngā taha e rua ki te 25.

$$x=\frac{1}{25}\left(-y+9\right)$$

Whakareatia \frac{1}{25} ki te -y+9.

$$x=-\frac{1}{25}y+\frac{9}{25}$$

Whakakapia te \frac{-y+9}{25} mō te x ki tērā atu whārite, 1.6x+0.2y=13.

$$1.6\left(-\frac{1}{25}y+\frac{9}{25}\right)+0.2y=13$$

Whakareatia 1.6 ki te \frac{-y+9}{25}.

$$-\frac{8}{125}y+\frac{72}{125}+0.2y=13$$

Tāpiri -\frac{8y}{125} ki te \frac{y}{5}.

$$\frac{17}{125}y+\frac{72}{125}=13$$

Me tango \frac{72}{125} mai i ngā taha e rua o te whārite.

$$\frac{17}{125}y=\frac{1553}{125}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{17}{125}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{1553}{17}$$

Whakaurua te \frac{1553}{17} mō y ki x=-\frac{1}{25}y+\frac{9}{25}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{25}\times \frac{1553}{17}+\frac{9}{25}$$

Whakareatia -\frac{1}{25} ki te \frac{1553}{17} 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.

$$x=-\frac{1553}{425}+\frac{9}{25}$$

Tāpiri \frac{9}{25} ki te -\frac{1553}{425} 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.

$$x=-\frac{56}{17}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{56}{17},y=\frac{1553}{17}$$

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.

$$25x+y=9,1.6x+0.2y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}25&1\\1.6&0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}25&1\\1.6&0.2\end{matrix}\right).

$$inverse(\left(\begin{matrix}25&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}25&1\\1.6&0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}25&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}9\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}25&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}9\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}25&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}9\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{0.2}{25\times 0.2-1.6}&-\frac{1}{25\times 0.2-1.6}\\-\frac{1.6}{25\times 0.2-1.6}&\frac{25}{25\times 0.2-1.6}\end{matrix}\right)\left(\begin{matrix}9\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{17}&-\frac{5}{17}\\-\frac{8}{17}&\frac{125}{17}\end{matrix}\right)\left(\begin{matrix}9\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{17}\times 9-\frac{5}{17}\times 13\\-\frac{8}{17}\times 9+\frac{125}{17}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{56}{17}\\\frac{1553}{17}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{56}{17},y=\frac{1553}{17}$$

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.

$$25x+y=9,1.6x+0.2y=13$$

Kia ōrite ai a 25x me \frac{8x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1.6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 25.

$$1.6\times 25x+1.6y=1.6\times 9,25\times 1.6x+25\times 0.2y=25\times 13$$

Whakarūnātia.

$$40x+1.6y=14.4,40x+5y=325$$

Me tango 40x+5y=325 mai i 40x+1.6y=14.4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$40x-40x+1.6y-5y=14.4-325$$

Tāpiri 40x ki te -40x. Ka whakakore atu ngā kupu 40x me -40x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$1.6y-5y=14.4-325$$

Tāpiri \frac{8y}{5} ki te -5y.

$$-3.4y=14.4-325$$

Tāpiri 14.4 ki te -325.

$$-3.4y=-310.6$$

Whakawehea ngā taha e rua o te whārite ki te -3.4, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{1553}{17}$$

Whakaurua te \frac{1553}{17} mō y ki 1.6x+0.2y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$1.6x+0.2\times \frac{1553}{17}=13$$

Whakareatia 0.2 ki te \frac{1553}{17} 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.

$$1.6x+\frac{1553}{85}=13$$

Me tango \frac{1553}{85} mai i ngā taha e rua o te whārite.

$$1.6x=-\frac{448}{85}$$

Whakawehea ngā taha e rua o te whārite ki te 1.6, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{56}{17}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{56}{17},y=\frac{1553}{17}$$