To solve this system of linear equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

Given equations:
1) 7x + 8y = 5
2) 3x + 5y = 17

To eliminate one of the variables, we can multiply the first equation by 5 and the second equation by 8:

5(7x + 8y) = 5(5)
8(3x + 5y) = 8(17)

This gives us:
35x + 40y = 25
24x + 40y = 136

Now, we can subtract the second equation from the first equation to eliminate y:

35x + 40y - 24x - 40y = 25 - 136
11x = -111
x = -111/11
x = -10

Now, substitute the value of x back into one of the original equations to solve for y. Let's use the first equation:

7x + 8y = 5
7(-10) + 8y = 5
-70 + 8y = 5
8y = 75
y = 75/8
y = 9.375

Therefore, the solution to the system of equations is x = -10 and y = 9.375.