i need to know when a linux distro was installed on the system

The above will be effective on reboot. With the Gnome Interface, when you click on a user name, the System name and the timestamp will appear while you enter the paassword.

#!/bin/bash
#
# This script generates a one liner file which is positioned on the
# Gnome Logon Screen above the first name on the list. The intention
# is to present a message to the user, prior to his logging 
# into the system.  However, this script serves to identify the system
# being logged into by using a field from /etc/os-release
# The script has been tested with SUSE, and Fedora (2 versions)
# It is GPL'd -- no license. Do with it what you want. I take 
# No responsibilities if the message is too long or too short.
# or the message is inappropriate.
#
# Version Sun Jan 16 18:33:29 
# Author Leslie Satenstein
# create a message on the logon screen.
#echo "Run as sudo $0"
#########################################################################
function TEST_FOR_ROOT_ACCESS                                           #
{                                                                       #
if [ $EUID -ne 0 ]; then                                                #
   echo ""                                                              #
   echo "##################################################"            #
   echo "#       You cannot run as user $LOGNAME           "            #
   echo "#       You cannot run as user $LOGNAME           "            #
   echo "#       You cannot run as user $LOGNAME           "            #
   echo "#                                 "        #
   echo "#       You must run as root                 "         #
   echo "#       You must run as root                 "         #
   echo "##################################################"            #
   echo ""                                                              #
   exit 1                                   #
fi                                                                      #      
}                                                                       #
#                                                                       #       
#########################################################################

TEST_FOR_ROOT_ACCESS        #root access ?
#setup vars
DIR1=/tmp/
DIR2=/tmp/
DCONF="echo"
#  ======================================================
#  If testing, comment out the following three lines 
#  or modify this script
#  and uncomment (if not done) the preceding three lines
#  ======================================================
DIR1=/etc/dconf/profile         #redefine
DIR2=/etc/dconf/db/gdm.d        #redefine
DCONF=dconf             #redefine
#
if [ ! -f $DIR1/gdm ]; then
# Following does the same as previous 3 with 1 line
echo -e "user-db:user\nsystem-db:gdm\nfile-db:/usr/share/gdm/greeter-dconf-defaults"  > $DIR1/gdm
fi

mkdir -p $DIR2

# Following are typical contents of /etc/dconf/db/gdm.d/01-banner-message

P1="banner-message-text='"
P2=$(grep PRETTY_NAME /etc/os-release |  cut -d\" -f2)
if [ -d /boot/lost+found ]; then
    P3=$(stat /boot/lost+found | grep ^Modify | cut -d' ' -f2-| cut -d'.' -f1)
else
    P3=$(stat /boot | grep ^Modify | cut -d' ' -f2-| cut -d'.' -f1)

fi

echo  "installing to $DIR1 and $DIR2"
echo  "[org/gnome/login-screen]"             >  ${DIR2}/01-banner-message
echo  "banner-message-enable=true"                      >>  ${DIR2}/01-banner-message
echo "$P1$P2 installed on\\r\\n$P3'"            >>  ${DIR2}/01-banner-message
echo "#Created by $0 at $(date)"            >>  ${DIR2}/01-banner-message
${DCONF} update
echo -e "$P2\ninstalled on: $P3"
exit 0**strong text**

If it is a new Linux system, you can look at the timestamp of /boot/lost+found.

If it is not there, for RedHat type systems try the timestamp of /etc/os-release

For example, my system was installed:

[root@scientific etc]# ls -l os-release -rw-r--r--. 1 root root 484 Feb 24 12:33 os-release