# How do you solve the following system?:  1/6x-3/5y=1 , 4/5x+2/3y = 2

Jul 10, 2018

$x = \frac{60}{19} , y = - \frac{15}{19}$

#### Explanation:

Multiply the first equation by 10 $\implies \frac{10}{6} x - 6 y = 10$

Multiply the second equation by 9 $\implies \frac{36}{5} x + 6 y = 18$

Add these two equations together to give $\frac{133}{15} x = 28$

Divide by $\frac{133}{15} \implies x = \frac{60}{19}$

Substitute this into the first equation

$\implies \frac{1}{6} \times \frac{60}{19} - \frac{3}{5} y = 1$

$\implies \frac{10}{19} - \frac{3}{5} y = 1$

$\implies - \frac{3}{5} y = \frac{9}{19}$

$\implies y = - \frac{15}{19}$

Jul 10, 2018

$x = \frac{60}{19} , y = - \frac{15}{19}$

#### Explanation:

Solving the first equation for $x$:

$\frac{1}{6} \cdot x = 1 + \frac{3}{5} \cdot y$
so
$x = 6 + \frac{18}{5} y$
substituting this in the second equation:

$\frac{4}{5} \cdot \left(6 + \frac{18}{5} y\right) + \frac{2}{3} y = 2$

expanding

$\frac{24}{5} + \frac{72}{25} y + \frac{2}{3} y = 2$
adding $- \frac{24}{5}$

$\frac{72}{25} y + \frac{2}{3} y = \frac{10 - 24}{5}$

note that $\frac{72}{25} + \frac{2}{3} = \frac{266}{75}$

so we get

$\frac{266}{75} y = - \frac{14}{5}$

multiplying by $\frac{75}{266}$

we get $y = - \frac{15}{19}$

so $x = 6 + \frac{18}{5} \cdot \left(- \frac{15}{19}\right)$

$x = \frac{60}{19}$