#### Answer

$x=-7$

#### Work Step by Step

We may substitute the expressions involving $x$ into the given equation $y_{1}-y_{2}=1$ and solve:
$(\frac{x-3}{5})-(\frac{x-5}{4})=1$
$(\frac{4}{4})(\frac{x-3}{5})-(\frac{5}{5})(\frac{x-5}{4})=1$ (We multiply each of the fractions by a number equal to 1 so that the denominator of each becomes the least common denominator.)
$(\frac{4x-12}{20})-(\frac{5x-25}{20})=1$
$\frac{(4x-12)-(5x-25)}{20}=1$
$\frac{4x-12-5x+25}{20}=1$ (Don't forget to distribute the negative.)
$\frac{-x+13}{20}=1$
$(20)(\frac{-x+13}{20})=(20)(1)$ (Multiply both sides by 20 to eliminate the denominator.)
$-x+13=20$
$-x=20-13$
$-x=7$
$x=-7$
Because we are dealing with a linear equality in one variable, there is only one answer.