Whakaoti i te {l}{8x+y=24}{x-2y=20}

$$\left. \begin{array} { l } { 8 x + y = 24 } \\ { x - 2 y = 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=-8$$

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.

$$8x+y=24,x-2y=20$$

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

$$8x+y=24$$

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

$$8x=-y+24$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-y+24\right)$$

Whakareatia \frac{1}{8} ki te -y+24.

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

Whakakapia te -\frac{y}{8}+3 mō te x ki tērā atu whārite, x-2y=20.

$$-\frac{1}{8}y+3-2y=20$$

Tāpiri -\frac{y}{8} ki te -2y.

$$-\frac{17}{8}y+3=20$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$-\frac{17}{8}y=17$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{17}{8}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-8$$

Whakaurua te -8 mō y ki x=-\frac{1}{8}y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{8}\left(-8\right)+3$$

Whakareatia -\frac{1}{8} ki te -8.

$$x=1+3$$

Tāpiri 3 ki te 1.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-8$$

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.

$$8x+y=24,x-2y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&1\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}24\\20\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}8&1\\1&-2\end{matrix}\right))\left(\begin{matrix}8&1\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\1&-2\end{matrix}\right))\left(\begin{matrix}24\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\1&-2\end{matrix}\right))\left(\begin{matrix}24\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\1&-2\end{matrix}\right))\left(\begin{matrix}24\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{8\left(-2\right)-1}&-\frac{1}{8\left(-2\right)-1}\\-\frac{1}{8\left(-2\right)-1}&\frac{8}{8\left(-2\right)-1}\end{matrix}\right)\left(\begin{matrix}24\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}&\frac{1}{17}\\\frac{1}{17}&-\frac{8}{17}\end{matrix}\right)\left(\begin{matrix}24\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}\times 24+\frac{1}{17}\times 20\\\frac{1}{17}\times 24-\frac{8}{17}\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=-8$$

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.

$$8x+y=24,x-2y=20$$

Kia ōrite ai a 8x 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 8.

$$8x+y=24,8x+8\left(-2\right)y=8\times 20$$

Whakarūnātia.

$$8x+y=24,8x-16y=160$$

Me tango 8x-16y=160 mai i 8x+y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+y+16y=24-160$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y+16y=24-160$$

Tāpiri y ki te 16y.

$$17y=24-160$$

Tāpiri 24 ki te -160.

$$17y=-136$$

Whakawehea ngā taha e rua ki te 17.

$$y=-8$$

Whakaurua te -8 mō y ki x-2y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-2\left(-8\right)=20$$

Whakareatia -2 ki te -8.

$$x+16=20$$

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

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-8$$