What is the value of a?
5
5 1/3
6 2/3
7

This is a neat little question. I don't think I've seen it before.
Step one 

Find c
3^2 + 4^2 = c^2
9 + 16 = c^2
c^2 = 25
c = 5

Step 2

Set up your first equation for b^2
a^2 + 4^2 = b^2 from triangle XWY

Step 3

Set up your second equation for b^2
25 +b^2 = (a + 3)^2 from triangle XWZ

Step 4

Put the results of Step 2 into step 3 and solve
25 + a^2 + 16 = (a + 3)^2 Collect the like terms on the left.
41 + a^2 = (a + 3)^2 Expand the brackets on the right
41 + a^2 = a^2 + 6a + 9 Transfer the 9 to the left.
32 + a^2 = a^2 + 6a Subtract a^2 from both sides.
6a = 32 Divide by 6
a = 32 / 6
a = 5 2/6
a = 5 1/3