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

Let's start by solving the first equation for x:

10x + 2y = 12
10x = 12 - 2y
x = (12 - 2y) / 10
x = (6 - y) / 5

Now, substitute this expression for x into the second equation to solve for y:

-5(6-y)/5 + 4y = -6
-6 + y + 4y = -6
5y = 0
y = 0

Now that we have found y, we can substitute it back into the first equation to find x:

10x + 2(0) = 12
10x = 12
x = 12 / 10
x = 1.2

Therefore, the solution to the system of equations is x = 1.2, y = 0.