I've Got A Hard Math Problem That Needs To Be Solved. Can You Help Please?

It is often easiest to formulate the equations for a problem of this nature if you assign a different variable to each of the quantities you want to determine, then write the equations according to the information in the problem statement.

We want to determine Arthur's weight. We will call Arthur's weight "A". We are given relations involving Lily's weight, too. We will call Lily's weight "L".

A = L+34    (Arthur weighs 34 pounds more than Lily)

A+L =    (their combined weight is ...)
   4L - 180    (180 pounds less than 4 times Lily's weight)

Since we want to find Arthur's weight (and we don't care that much about Lily's weight), we want to get rid of L in the second equation. We do that by using the first equation to write an expression for L.
L = A - 34    (we subtract 34 from both sides of the first equation)

Now, we substitute this into the second equation.
A + (A-34) = 4(A-34) - 180
2A - 34 = 4A - 136 - 180    (use the distributive property)
-34 = 2A - 316    (subtract 2A from both sides)
316 - 34 = 2A    (add 316 to both sides)
282/2 = A    (divide both sides by 2)
141 = A

Arthur weighs 141 pounds.

Check
To do a proper check, we can compute Lily's weight and see if the two weights meet the requirements.
L = A - 34 = 141 - 34 = 107
A+L = 4L - 180
141 + 107 = 4(107) - 180
248 = 428 - 180    (yes)

Let X represent Arthur's weight
Let Y represent Lily's weight

X+Y=180....1
Y+34=X....2

Sub X into 1
(Y+34)+Y=180
2Y=146
Y=73

Sub Y into 1
X+73=180
X=117

Therefore Arthur weighs 117 pounds.