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

Let's use the substitution method.

From the first equation, we can solve for y in terms of x:

4x + y = 3
y = 3 - 4x

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

6x - 2(3 - 4x) = 1
6x - 6 + 8x = 1
14x - 6 = 1
14x = 7
x = 7/14
x = 1/2

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

4(1/2) + y = 3
2 + y = 3
y = 1

Therefore, the solution to the system of equations is x = 1/2 and y = 1.