Whakaoti i te {l}{40x+4y=80}{8x-15y=-1}

$$\left. \begin{array} { l } { 40 x + 4 y = 80 } \\ { 8 x - 15 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{299}{158} = 1\frac{141}{158} \approx 1.892405063$$

$$y = \frac{85}{79} = 1\frac{6}{79} \approx 1.075949367$$

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.

$$40x+4y=80,8x-15y=-1$$

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

$$40x+4y=80$$

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

$$40x=-4y+80$$

Whakawehea ngā taha e rua ki te 40.

$$x=\frac{1}{40}\left(-4y+80\right)$$

Whakareatia \frac{1}{40} ki te -4y+80.

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

Whakakapia te -\frac{y}{10}+2 mō te x ki tērā atu whārite, 8x-15y=-1.

$$8\left(-\frac{1}{10}y+2\right)-15y=-1$$

Whakareatia 8 ki te -\frac{y}{10}+2.

$$-\frac{4}{5}y+16-15y=-1$$

Tāpiri -\frac{4y}{5} ki te -15y.

$$-\frac{79}{5}y+16=-1$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$-\frac{79}{5}y=-17$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{79}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{85}{79}$$

Whakaurua te \frac{85}{79} mō y ki x=-\frac{1}{10}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{10}\times \frac{85}{79}+2$$

Whakareatia -\frac{1}{10} ki te \frac{85}{79} 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.

$$x=-\frac{17}{158}+2$$

Tāpiri 2 ki te -\frac{17}{158}.

$$x=\frac{299}{158}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{299}{158},y=\frac{85}{79}$$

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.

$$40x+4y=80,8x-15y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}40&4\\8&-15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}80\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}40&4\\8&-15\end{matrix}\right).

$$inverse(\left(\begin{matrix}40&4\\8&-15\end{matrix}\right))\left(\begin{matrix}40&4\\8&-15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&4\\8&-15\end{matrix}\right))\left(\begin{matrix}80\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&4\\8&-15\end{matrix}\right))\left(\begin{matrix}80\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&4\\8&-15\end{matrix}\right))\left(\begin{matrix}80\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{40\left(-15\right)-4\times 8}&-\frac{4}{40\left(-15\right)-4\times 8}\\-\frac{8}{40\left(-15\right)-4\times 8}&\frac{40}{40\left(-15\right)-4\times 8}\end{matrix}\right)\left(\begin{matrix}80\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{632}&\frac{1}{158}\\\frac{1}{79}&-\frac{5}{79}\end{matrix}\right)\left(\begin{matrix}80\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{632}\times 80+\frac{1}{158}\left(-1\right)\\\frac{1}{79}\times 80-\frac{5}{79}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{299}{158}\\\frac{85}{79}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{299}{158},y=\frac{85}{79}$$

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.

$$40x+4y=80,8x-15y=-1$$

Kia ōrite ai a 40x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 40.

$$8\times 40x+8\times 4y=8\times 80,40\times 8x+40\left(-15\right)y=40\left(-1\right)$$

Whakarūnātia.

$$320x+32y=640,320x-600y=-40$$

Me tango 320x-600y=-40 mai i 320x+32y=640 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$320x-320x+32y+600y=640+40$$

Tāpiri 320x ki te -320x. Ka whakakore atu ngā kupu 320x me -320x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$32y+600y=640+40$$

Tāpiri 32y ki te 600y.

$$632y=640+40$$

Tāpiri 640 ki te 40.

$$632y=680$$

Whakawehea ngā taha e rua ki te 632.

$$y=\frac{85}{79}$$

Whakaurua te \frac{85}{79} mō y ki 8x-15y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x-15\times \frac{85}{79}=-1$$

Whakareatia -15 ki te \frac{85}{79}.

$$8x-\frac{1275}{79}=-1$$

Me tāpiri \frac{1275}{79} ki ngā taha e rua o te whārite.

$$8x=\frac{1196}{79}$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{299}{158}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{299}{158},y=\frac{85}{79}$$