Whakaoti i te {l}{9x+8y=5280x?}{8x+12y=5280*9}

$$\left. \begin{array} { l } { 9 x + 8 y = 5280 x ? } \\ { 8 x + 12 y = 5280 \times 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{8640}{1439} = 6\frac{6}{1439} \approx 6.004169562$$

$$y = \frac{5692680}{1439} = 3955\frac{1435}{1439} \approx 3955.997220292$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 5280x mai i ngā taha e rua.

$$9x+8y-5280x=0$$

Pahekotia te 9x me -5280x, ka -5271x.

$$-5271x+8y=0$$

Whakaarohia te whārite tuarua. Whakareatia te 5280 ki te 9, ka 47520.

$$8x+12y=47520$$

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.

$$-5271x+8y=0,8x+12y=47520$$

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

$$-5271x+8y=0$$

Me tango 8y mai i ngā taha e rua o te whārite.

$$-5271x=-8y$$

Whakawehea ngā taha e rua ki te -5271.

$$x=-\frac{1}{5271}\left(-8\right)y$$

Whakareatia -\frac{1}{5271} ki te -8y.

$$x=\frac{8}{5271}y$$

Whakakapia te \frac{8y}{5271} mō te x ki tērā atu whārite, 8x+12y=47520.

$$8\times \frac{8}{5271}y+12y=47520$$

Whakareatia 8 ki te \frac{8y}{5271}.

$$\frac{64}{5271}y+12y=47520$$

Tāpiri \frac{64y}{5271} ki te 12y.

$$\frac{63316}{5271}y=47520$$

Whakawehea ngā taha e rua o te whārite ki te \frac{63316}{5271}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{5692680}{1439}$$

Whakaurua te \frac{5692680}{1439} mō y ki x=\frac{8}{5271}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{8}{5271}\times \frac{5692680}{1439}$$

Whakareatia \frac{8}{5271} ki te \frac{5692680}{1439} 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.

$$x=\frac{8640}{1439}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8640}{1439},y=\frac{5692680}{1439}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 5280x mai i ngā taha e rua.

$$9x+8y-5280x=0$$

Pahekotia te 9x me -5280x, ka -5271x.

$$-5271x+8y=0$$

Whakaarohia te whārite tuarua. Whakareatia te 5280 ki te 9, ka 47520.

$$8x+12y=47520$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$-5271x+8y=0,8x+12y=47520$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\47520\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5271&8\\8&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right))\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right))\left(\begin{matrix}0\\47520\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right))\left(\begin{matrix}0\\47520\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right))\left(\begin{matrix}0\\47520\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{-5271\times 12-8\times 8}&-\frac{8}{-5271\times 12-8\times 8}\\-\frac{8}{-5271\times 12-8\times 8}&-\frac{5271}{-5271\times 12-8\times 8}\end{matrix}\right)\left(\begin{matrix}0\\47520\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{15829}&\frac{2}{15829}\\\frac{2}{15829}&\frac{5271}{63316}\end{matrix}\right)\left(\begin{matrix}0\\47520\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{15829}\times 47520\\\frac{5271}{63316}\times 47520\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8640}{1439}\\\frac{5692680}{1439}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{8640}{1439},y=\frac{5692680}{1439}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 5280x mai i ngā taha e rua.

$$9x+8y-5280x=0$$

Pahekotia te 9x me -5280x, ka -5271x.

$$-5271x+8y=0$$

Whakaarohia te whārite tuarua. Whakareatia te 5280 ki te 9, ka 47520.

$$8x+12y=47520$$

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.

$$-5271x+8y=0,8x+12y=47520$$

Kia ōrite ai a -5271x 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 -5271.

$$8\left(-5271\right)x+8\times 8y=0,-5271\times 8x-5271\times 12y=-5271\times 47520$$

Whakarūnātia.

$$-42168x+64y=0,-42168x-63252y=-250477920$$

Me tango -42168x-63252y=-250477920 mai i -42168x+64y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-42168x+42168x+64y+63252y=250477920$$

Tāpiri -42168x ki te 42168x. Ka whakakore atu ngā kupu -42168x me 42168x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$64y+63252y=250477920$$

Tāpiri 64y ki te 63252y.

$$63316y=250477920$$

Whakawehea ngā taha e rua ki te 63316.

$$y=\frac{5692680}{1439}$$

Whakaurua te \frac{5692680}{1439} mō y ki 8x+12y=47520. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x+12\times \frac{5692680}{1439}=47520$$

Whakareatia 12 ki te \frac{5692680}{1439}.

$$8x+\frac{68312160}{1439}=47520$$

Me tango \frac{68312160}{1439} mai i ngā taha e rua o te whārite.

$$8x=\frac{69120}{1439}$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{8640}{1439}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8640}{1439},y=\frac{5692680}{1439}$$