Whakaoti i te {l}{3x+6y=210}{x/4+y/5=sqrt{210}}

$$\left. \begin{array} { l } { 3 x + 6 y = 210 } \\ { \frac { x } { 4 } + \frac { y } { 5 } = \sqrt { 210 } } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{20 \sqrt{210} - 140}{3} \approx 49.942511641$$

$$y = \frac{175 - 10 \sqrt{210}}{3} \approx 10.028744179$$

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.

$$3x+6y=210,\frac{1}{4}x+\frac{1}{5}y=\sqrt{210}$$

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

$$3x+6y=210$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$3x=-6y+210$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-6y+210\right)$$

Whakareatia \frac{1}{3} ki te -6y+210.

$$x=-2y+70$$

Whakakapia te -2y+70 mō te x ki tērā atu whārite, \frac{1}{4}x+\frac{1}{5}y=\sqrt{210}.

$$\frac{1}{4}\left(-2y+70\right)+\frac{1}{5}y=\sqrt{210}$$

Whakareatia \frac{1}{4} ki te -2y+70.

$$-\frac{1}{2}y+\frac{35}{2}+\frac{1}{5}y=\sqrt{210}$$

Tāpiri -\frac{y}{2} ki te \frac{y}{5}.

$$-\frac{3}{10}y+\frac{35}{2}=\sqrt{210}$$

Me tango \frac{35}{2} mai i ngā taha e rua o te whārite.

$$-\frac{3}{10}y=\sqrt{210}-\frac{35}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{10}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{175-10\sqrt{210}}{3}$$

Whakaurua te \frac{-10\sqrt{210}+175}{3} mō y ki x=-2y+70. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{175-10\sqrt{210}}{3}+70$$

Whakareatia -2 ki te \frac{-10\sqrt{210}+175}{3}.

$$x=\frac{20\sqrt{210}-350}{3}+70$$

Tāpiri 70 ki te \frac{20\sqrt{210}-350}{3}.

$$x=\frac{20\sqrt{210}-140}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{20\sqrt{210}-140}{3},y=\frac{175-10\sqrt{210}}{3}$$

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.

$$3x+6y=210,\frac{1}{4}x+\frac{1}{5}y=\sqrt{210}$$

Kia ōrite ai a 3x me \frac{x}{4}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{1}{4} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$\frac{1}{4}\times 3x+\frac{1}{4}\times 6y=\frac{1}{4}\times 210,3\times \frac{1}{4}x+3\times \frac{1}{5}y=3\sqrt{210}$$

Whakarūnātia.

$$\frac{3}{4}x+\frac{3}{2}y=\frac{105}{2},\frac{3}{4}x+\frac{3}{5}y=3\sqrt{210}$$

Me tango \frac{3}{4}x+\frac{3}{5}y=3\sqrt{210} mai i \frac{3}{4}x+\frac{3}{2}y=\frac{105}{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{3}{4}x-\frac{3}{4}x+\frac{3}{2}y-\frac{3}{5}y=\frac{105}{2}-3\sqrt{210}$$

Tāpiri \frac{3x}{4} ki te -\frac{3x}{4}. Ka whakakore atu ngā kupu \frac{3x}{4} me -\frac{3x}{4}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{3}{2}y-\frac{3}{5}y=\frac{105}{2}-3\sqrt{210}$$

Tāpiri \frac{3y}{2} ki te -\frac{3y}{5}.

$$\frac{9}{10}y=\frac{105}{2}-3\sqrt{210}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{10}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{175-10\sqrt{210}}{3}$$

Whakaurua te \frac{175-10\sqrt{210}}{3} mō y ki \frac{1}{4}x+\frac{1}{5}y=\sqrt{210}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{1}{4}x+\frac{1}{5}\times \frac{175-10\sqrt{210}}{3}=\sqrt{210}$$

Whakareatia \frac{1}{5} ki te \frac{175-10\sqrt{210}}{3}.

$$\frac{1}{4}x+\frac{35-2\sqrt{210}}{3}=\sqrt{210}$$

Me tango \frac{-2\sqrt{210}+35}{3} mai i ngā taha e rua o te whārite.

$$\frac{1}{4}x=\frac{5\sqrt{210}-35}{3}$$

Me whakarea ngā taha e rua ki te 4.

$$x=\frac{20\sqrt{210}-140}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{20\sqrt{210}-140}{3},y=\frac{175-10\sqrt{210}}{3}$$