#### Answer

Our solution is:
$$(2, 1)$$

#### Work Step by Step

To solve a system of equations by the addition method, you have to transform the equations so that the coefficients of one of the variables for both equations differ only in sign. In this problem, we want to work with the $y$ variable. We need to transform both equations.
We multiply the first equation by $-5$ and the second equation by $2$ to get:
$$-5(5x - 2y) = -5(8)$$
$$ 2(3x - 5y) = 2(1)$$
We distribute to get:
$$-25x + 10y = -40$$
$$ 6x - 10y = 2$$
We add them to get the following equation:
$$-19x = -38$$
Divide by $-19$ to solve for $x$:
$$x = 2$$
We can now plug in what we got for $x$ into one of the equations to get $y$:
$$5(2) - 2y = 8$$
Multiply, according to order of operations:
$$10 - 2y = 8$$
Subtract $7$ from each side to isolate the $x$ term:
$$-2y = -2$$
Solve for $x$ by dividing by $-2$ on both sides:
$$y = 1$$
Our solution is:
$$(2, 1)$$