Let's see here

let the width of the walk be x. since the walk surrounds the garden, the width of the garden is going to be it's width + 2x and the length of the garden is going to be it's length + 2x.

now just plug these values into the area equation:

we have length = 12 + 2x

we have width = 10 + 2x

area = length*width

=> area = 168 = (12 + 2x)(10 + 2x)

=> (12 + 2x)(10 + 2x) = 168

=> 120 + 44x + 4x^2 = 168 ............i expanded the brackets

=> 30 + 11x + x^2 = 42 .................i divided through by 4

=> x^2 + 11x - 12 = 0

=> (x + 12)(x - 1) = 0

=> x + 12 = 0, or x - 1 = 0

=> x = -12 or x = 1

x cannot be minus 12, since it's a length

so x = 1 ft

so the width of the walk is 1 ft