# How do you solve (5x-10)/(7x+6)=10/8?

Aug 5, 2016

I got $x = - \frac{14}{3}$.

I would first multiply by $8$, and then by $7 x + 6$.

$\frac{5 x - 10}{7 x + 6} = \frac{10}{8}$

$\frac{8 \cdot \left(5 x - 10\right)}{7 x + 6} = \frac{10}{\cancel{8}} \cdot \cancel{8}$

$\frac{8 \left(5 x - 10\right)}{\cancel{\left(7 x + 6\right)}} \cdot \cancel{\left(7 x + 6\right)} = 10 \cdot \left(7 x + 6\right)$

From here, just distribute the terms, move the same types of terms to each side, and solve for $x$.

$\underline{8} \left(\underline{\textcolor{red}{5}} x - \underline{\textcolor{\mathrm{da} r k b l u e}{10}}\right) = \underline{10} \left(\underline{\textcolor{red}{7}} x + \underline{\textcolor{\mathrm{da} r k b l u e}{6}}\right)$

$\textcolor{red}{40} x - \textcolor{\mathrm{da} r k b l u e}{80} = \textcolor{red}{70} x + \textcolor{\mathrm{da} r k b l u e}{60}$

$- 140 = 30 x$

$\textcolor{b l u e}{x} = - \frac{140}{30}$

$= \textcolor{b l u e}{- \frac{14}{3}}$

And we can prove that this is correct:

(5(-14/3) - 10)/(7(-14/3) + 6) stackrel(?" ")(=) 10/8

(-70/3 - 30/3)/(-98/3 + 18/3) stackrel(?" ")(=) 10/8

(-100/3)/(-80/3) stackrel(?" ")(=) 10/8

color(red)(cancel(color(black)(-)))100/color(red)(cancel(color(black)(3)))*color(red)(cancel(color(black)(-)))color(red)(cancel(color(black)(3)))/80 stackrel(?" ")(=) 10/8

$\frac{100}{80} = \textcolor{g r e e n}{\frac{10}{8} = \frac{10}{8}}$