to find the slope, change the equation to slope-intercept form
y = mx + b
the m is the slope and the b is the y-intercept
so...
x + 4y = 6
subtract x from both sides
4y = -x + 6
isolate y by dividing both sides by 4
y = -1/4x + 6/4
simplify
y = -1/4x + 3/2
compare
y = mx + b
m is the slope
so -1/4 is the slope of this equation
Hope this helps :)