Whakaoti i te {l}{8x+2y=104}{1x+1y=25}

$$\left. \begin{array} { l } { 8 x + 2 y = 104 } \\ { 1 x + 1 y = 25 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=16$$

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.

$$8x+2y=104,x+y=25$$

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

$$8x+2y=104$$

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

$$8x=-2y+104$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-2y+104\right)$$

Whakareatia \frac{1}{8} ki te -2y+104.

$$x=-\frac{1}{4}y+13$$

Whakakapia te -\frac{y}{4}+13 mō te x ki tērā atu whārite, x+y=25.

$$-\frac{1}{4}y+13+y=25$$

Tāpiri -\frac{y}{4} ki te y.

$$\frac{3}{4}y+13=25$$

Me tango 13 mai i ngā taha e rua o te whārite.

$$\frac{3}{4}y=12$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=16$$

Whakaurua te 16 mō y ki x=-\frac{1}{4}y+13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{4}\times 16+13$$

Whakareatia -\frac{1}{4} ki te 16.

$$x=-4+13$$

Tāpiri 13 ki te -4.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=16$$

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.

$$8x+2y=104,x+y=25$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&2\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}104\\25\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}8&2\\1&1\end{matrix}\right))\left(\begin{matrix}8&2\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\1&1\end{matrix}\right))\left(\begin{matrix}104\\25\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\1&1\end{matrix}\right))\left(\begin{matrix}104\\25\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\1&1\end{matrix}\right))\left(\begin{matrix}104\\25\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8-2}&-\frac{2}{8-2}\\-\frac{1}{8-2}&\frac{8}{8-2}\end{matrix}\right)\left(\begin{matrix}104\\25\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&-\frac{1}{3}\\-\frac{1}{6}&\frac{4}{3}\end{matrix}\right)\left(\begin{matrix}104\\25\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 104-\frac{1}{3}\times 25\\-\frac{1}{6}\times 104+\frac{4}{3}\times 25\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\16\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=16$$

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.

$$8x+2y=104,x+y=25$$

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.

$$8x+2y=104,8x+8y=8\times 25$$

Whakarūnātia.

$$8x+2y=104,8x+8y=200$$

Me tango 8x+8y=200 mai i 8x+2y=104 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+2y-8y=104-200$$

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.

$$2y-8y=104-200$$

Tāpiri 2y ki te -8y.

$$-6y=104-200$$

Tāpiri 104 ki te -200.

$$-6y=-96$$

Whakawehea ngā taha e rua ki te -6.

$$y=16$$

Whakaurua te 16 mō y ki x+y=25. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+16=25$$

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

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=16$$