Lahenda sqrt{3/5}(x+1)+sqrt{5/3}(x-1)=1/15

$$\sqrt { \frac { 3 } { 5 } } ( x + 1 ) + \sqrt { \frac { 5 } { 3 } } ( x - 1 ) = \frac { 1 } { 15 }$$

Lahendage ja leidke x

$$x=\frac{\sqrt{15}+30}{120}\approx 0,282274861$$

Juhised lineaarvõrrandi lahendamiseks

Kirjutage: allüksus \sqrt{\frac{3}{5}}: allüksus juured \frac{\sqrt{3}}{\sqrt{5}}.

$$\frac{\sqrt{3}}{\sqrt{5}}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Ratsionaliseerige korrutades lugeja ja \sqrt{5} nimetaja \frac{\sqrt{3}}{\sqrt{5}} nimetaja.

$$\frac{\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

\sqrt{5} ruut on 5.

$$\frac{\sqrt{3}\sqrt{5}}{5}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

\sqrt{3} ja \sqrt{5} korrutage numbrid, mis on sama juur.

$$\frac{\sqrt{15}}{5}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Avaldage \frac{\sqrt{15}}{5}\left(x+1\right) ühe murdarvuna.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Kirjutage: allüksus \sqrt{\frac{5}{3}}: allüksus juured \frac{\sqrt{5}}{\sqrt{3}}.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{5}}{\sqrt{3}}\left(x-1\right)=\frac{1}{15}$$

Ratsionaliseerige korrutades lugeja ja \sqrt{3} nimetaja \frac{\sqrt{5}}{\sqrt{3}} nimetaja.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\left(x-1\right)=\frac{1}{15}$$

\sqrt{3} ruut on 3.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{5}\sqrt{3}}{3}\left(x-1\right)=\frac{1}{15}$$

\sqrt{5} ja \sqrt{3} korrutage numbrid, mis on sama juur.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{15}}{3}\left(x-1\right)=\frac{1}{15}$$

Avaldage \frac{\sqrt{15}}{3}\left(x-1\right) ühe murdarvuna.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{15}\left(x-1\right)}{3}=\frac{1}{15}$$

Avaldiste liitmiseks või lahutamiseks laiendage need, et neil oleksid ühised nimetajad. 5 ja 3 vähim ühiskordne on 15. Korrutage omavahel \frac{\sqrt{15}\left(x+1\right)}{5} ja \frac{3}{3}. Korrutage omavahel \frac{\sqrt{15}\left(x-1\right)}{3} ja \frac{5}{5}.

$$\frac{3\sqrt{15}\left(x+1\right)}{15}+\frac{5\sqrt{15}\left(x-1\right)}{15}=\frac{1}{15}$$

Kuna murdudel \frac{3\sqrt{15}\left(x+1\right)}{15} ja \frac{5\sqrt{15}\left(x-1\right)}{15} on sama nimetaja, liitke nende lugejad.

$$\frac{3\sqrt{15}\left(x+1\right)+5\sqrt{15}\left(x-1\right)}{15}=\frac{1}{15}$$

Tehke korrutustehted võrrandis 3\sqrt{15}\left(x+1\right)+5\sqrt{15}\left(x-1\right).

$$\frac{3\sqrt{15}x+3\sqrt{15}+5\sqrt{15}x-5\sqrt{15}}{15}=\frac{1}{15}$$

Kombineerige sarnased liikmed avaldises 3\sqrt{15}x+3\sqrt{15}+5\sqrt{15}x-5\sqrt{15}.

$$\frac{8\sqrt{15}x-2\sqrt{15}}{15}=\frac{1}{15}$$

Korrutage mõlemad pooled 15-ga.

$$8\sqrt{15}x-2\sqrt{15}=\frac{1}{15}\times 15$$

Taandage 15 ja 15.

$$8\sqrt{15}x-2\sqrt{15}=1$$

Liitke 2\sqrt{15} mõlemale poolele.

$$8\sqrt{15}x=1+2\sqrt{15}$$

Võrrand on standardkujul.

$$8\sqrt{15}x=2\sqrt{15}+1$$

Jagage mõlemad pooled 8\sqrt{15}-ga.

$$\frac{8\sqrt{15}x}{8\sqrt{15}}=\frac{2\sqrt{15}+1}{8\sqrt{15}}$$

8\sqrt{15}-ga jagamine võtab 8\sqrt{15}-ga korrutamise tagasi.

$$x=\frac{2\sqrt{15}+1}{8\sqrt{15}}$$

Jagage 1+2\sqrt{15} väärtusega 8\sqrt{15}.

$$x=\frac{\sqrt{15}}{120}+\frac{1}{4}$$