"She's kinda got a little age on her Captain" (Scottie / Star Trek)
The computer is old enough that I'm betting the Power Supply is bad. However could just be the Power On switch.
Gonna' be a long read, better put the coffee pot on.
There is a simple test to see if the problem is a $5 Power On switch, or the Power Supply.
Involves bypassing the Power On switch.
Does NOT involve the wires of the Power On switch, nor the switch itself, however.
The Power Supply of your computer is a converter. It converts 100 to 240 Volts AC, to three low DC voltages;
A) 3.3 Volts DC
B) 5 Volts DC
C) 12 Volts DC
In comparison two D cell flashlight batteries produce 3 Volts DC.
Stated in case there was concern about getting shocked.
Let's go to HP Support;
HP Support > Pavilion a735w Desktop PC > Main Support page,http://h10025.www1.hp.com/ewfrf/wc/product?product=435559&lc=en&cc=us&dlc=en&lang=en&cc=us
A look at the motherboard; (No hardware installed, Top View),
Product information > Product specifications > Motherboard Specifications, A7V8X-LA (Kelut),http://h10025.www1.hp.com/ewfrf/wc/document?docname=c00069442&tmp_task=prodinfoCategory&cc=us&dlc=en&lang=en&lc=en&product=435559
Scroll down to the motherboard layout illustration, and motherboard photo.
In the layout illustration look at the right side. Look at the vertical (Rectangular) 'box' with ATX Power Connector in it.
Looking at the motherboard photo you will note, it is a whitish connector with two columns of 10 socket holes.
You will see it has a 20-pin ATX main power cable in it, like this general example, (NOTE* Color does NOT matter for connectors),http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain20
The ATX main power cable will be plugged in, as shown by the photo to the far right.
A jumper wire is used to bypass the Power On switch.
The preferred jumper wire is a straightened out paperclip, bent into a U-shape.
The top of the U, is wrapped a few times with black plastic electrical tape. This taped area is for your fingers, and thumb to hold onto.
Turn the U over, and it is the 'legs' of the U that you are going to use.
The BACK of the ATX main power cable's connector, is where the wires go in.
This is where the U-shaped jumper wire goes also.
One 'leg' goes down in a socket hole, RIGHT NEXT TO the existing wire that is in the socket hole; and the GREEN wire.
The Green wire is the Soft Power On wire.
The leg slides down right next to the insulation of the wire, and touches a metal terminal, at the end of the wire.
This is an example of the metal terminal,http://www.molex.com/molex/products/datasheet.jsp?part=active/0002081202_CRIMP_TERMINALS.xml&channel=Products&Lang=en-US
(Left side of metal terminal shown, is crimped around the insulation of the wire. This is what the U-shaped jumper wire touches. The right side shown is partially visible, in the center photo of the above Playtool link. Just the tip openings are shown. Like sort of like brass O shapes )
The leg of the U-shaped jumper wire has to go pretty far down in the socket, to touch that metal terminal. The jumper wire MUST touch that metal terminal. (About 3/8ths of an Inch, or about 9.5mm)
The other leg of the jumper wire, goes down into ANY socket hole that has a Black wire in it. ALL Black wires are Ground wires. (Negative)
Power Supply (Computer) plugged into power, use the U-shaped jumper wire.
(Bend to fit before plugging computer into power, to test fit first.
Not stating that there may be a small spark, as you connect.
That's the way electricity works. Use a D cell flashlight battery, flashlight bulb, and a small wire. There is a small spark when you connect.
For this reason you may also wish to wear a glove, on the 'jumper wire hand' )
Using the jumper wire, and;
1) Computer (Power Supply) comes on, you have a bad Power On switch.
2) Computer (Power Supply) does NOT come on, you have a bad Power Supply.
The two leading causes of desktop computer failure is;
Computer is dirty inside
Bad Power Supply
How to open the computer case,
Main Support page > How to > Adding / Replacing hardware > Opening the PC case,http://h10025.www1.hp.com/ewfrf/wc/document?docname=c00593536&tmp_task=useCategory&cc=us&dlc=en&lang=en&lc=en&product=435559http://h20000.www2.hp.com/bizsupport/TechSupport/Document.jsp?lang=en&cc=us&prodSeriesId=1126422&objectID=c01358091
(Watch out! Video has volume turned ALL the way up! You may wish to turn your volume down, before clicking on the above link)http://www.fonerbooks.com/r_power.htm
Make a drawing/s, and notes as to where the wires go, before removing the Power Supply.
Especially the Front Panel header wires.
The Front Panel is the plastic bezel on the front of your computer.
The area of contact pins on the motherboard, that the main wires from the Front Panel go to; is the Front Panel header.
Sometimes one, or more wires may accidentally be popped off.
FOLLOW Anti-Static Precautions.
Your body carries Static electricity. Static WILL fry out, (Short Circuit), the delicate hardware components inside a computer
Relieve your body of Static, BEFORE reaching inside your unplugged from power, computer.
Computer on a table, computer Unplugged from power, computer case open;
TOUCH an unpainted surface, of the metal frame of the open computer case.
This action will relieve your body of Static.
IF you leave your computer in the middle of working on it, be SURE to Touch the metal frame again, upon your return.
(Don't forget to plug back into power, before using the U-shaped jumper wire)
The Power On switch is an ATX power on switch, like this general example,http://www.directron.com/atxswitch.html
The Power On switch is located inside the plastic housing, of the Power On Button assembly.
You have to remove the Front Panel, to access it.
HP Support main page above > How To > Adding/Replacing hardware > Removing the Front Cover
Power On Button assembly removed, (Make sure you note, which pins in the Front Panel header on the motherboard, those Power On switch wires connect to), I use a hair dryer to soften the plastic.
Soften the plastic of the Power On Button assembly.
It gets brittle with age, and heat.
I soften it enough to ease a tab out a little, and maybe the opposite tab, and ease the old Power On switch out.
You'll see what I mean, once you have the Power On Button assembly in your hand.
Tab on Power On Button assembly broke, no matter how much finesse you used? Happens. I 'dot it' with a small blob of hot glue, for a hot glue gun.
(When finished after replacing a Power Supply, take the Ram Memory module/s out ('Stick'), and reinstall. They always seem to get bumped.
Visually inspecting will NOT do. You have to remove, and reinstall; to be assured they/it are installed correctly, and tightly.
May wish to do the same after replacing a Power On switch, also)
For additional questions please post in a Comment.