Issolvi frac{5+2sqrt{3}}{7+4sqrt{3}}=x+sqrt{3}y

$$\frac { 5 + 2 \sqrt { 3 } } { 7 + 4 \sqrt { 3 } } = x + \sqrt { 3 } y$$

Solvi għal y

$$y=-\frac{\sqrt{3}\left(x+6\sqrt{3}-11\right)}{3}$$

Passi biex Issolvi l-Ekwazzjoni Lineari

Irrazzjonalizza d-denominatur tal-\frac{5+2\sqrt{3}}{7+4\sqrt{3}} billi timmultiplika in-numeratur u d-denominatur mill-7-4\sqrt{3}.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}=x+\sqrt{3}y$$

Ikkunsidra li \left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{7^{2}-\left(4\sqrt{3}\right)^{2}}=x+\sqrt{3}y$$

Ikkalkula 7 bil-power ta' 2 u tikseb 49.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-\left(4\sqrt{3}\right)^{2}}=x+\sqrt{3}y$$

Espandi \left(4\sqrt{3}\right)^{2}.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-4^{2}\left(\sqrt{3}\right)^{2}}=x+\sqrt{3}y$$

Ikkalkula 4 bil-power ta' 2 u tikseb 16.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\left(\sqrt{3}\right)^{2}}=x+\sqrt{3}y$$

Il-kwadrat ta' \sqrt{3} huwa 3.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\times 3}=x+\sqrt{3}y$$

Immultiplika 16 u 3 biex tikseb 48.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-48}=x+\sqrt{3}y$$

Naqqas 48 minn 49 biex tikseb 1.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{1}=x+\sqrt{3}y$$

Kwalunkwe ħaġa diviża b'wieħed tagħti riżultat tagħha stess.

$$\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)=x+\sqrt{3}y$$

Uża l-propjetà distributtiva biex timmultiplika 5+2\sqrt{3} b'7-4\sqrt{3} u kkombina termini simili.

$$35-6\sqrt{3}-8\left(\sqrt{3}\right)^{2}=x+\sqrt{3}y$$

Il-kwadrat ta' \sqrt{3} huwa 3.

$$35-6\sqrt{3}-8\times 3=x+\sqrt{3}y$$

Immultiplika -8 u 3 biex tikseb -24.

$$35-6\sqrt{3}-24=x+\sqrt{3}y$$

Naqqas 24 minn 35 biex tikseb 11.

$$11-6\sqrt{3}=x+\sqrt{3}y$$

Ibdel in-naħat sabiex it-termini varjabbli kollha jkunu fuq in-naħa tax-xellug.

$$x+\sqrt{3}y=11-6\sqrt{3}$$

Naqqas x miż-żewġ naħat.

$$\sqrt{3}y=11-6\sqrt{3}-x$$

L-ekwazzjoni hija f'forma standard.

$$\sqrt{3}y=-x+11-6\sqrt{3}$$

Iddividi ż-żewġ naħat b'\sqrt{3}.

$$\frac{\sqrt{3}y}{\sqrt{3}}=\frac{-x+11-6\sqrt{3}}{\sqrt{3}}$$

Meta tiddividi b'\sqrt{3} titneħħa l-multiplikazzjoni b'\sqrt{3}.

$$y=\frac{-x+11-6\sqrt{3}}{\sqrt{3}}$$

Iddividi -6\sqrt{3}-x+11 b'\sqrt{3}.

$$y=\frac{\sqrt{3}\left(-x+11-6\sqrt{3}\right)}{3}$$

Solvi għal x

$$x=-\sqrt{3}y+11-6\sqrt{3}$$