That's where you're wrong. Google will correct you just by doing some searching. They ended the "Border of your highest ranking across all modes" in Season 2. With the wake of Season 3, you now only receive the border trim of your highest achievement in that particular mode. If you're Plat in Solo Q, but Bronze in 5v5's, you will have a Platinum Solo q border, but no border for 5v5's unless you reach Silver, Gold, Plat, etc. in 5v5's. So, no.. If you start a team with 4 others and do your 5 placement matches, even winning all 5, you will only get a Silver border. (Community feedback)
**Edit: Riot did this for the simple fact that, if a Bronze player has friends in Diamond or Challenger for instance, and they jump on their Diamond/Chall 5's team, and take part in 30% games or 10 wins. They would receive a Diamond border in ranked solo and 5's. And if they were to receive that border for Solo Q, then magically they'd have an ELO boost as well in Solo Q. Instead, the only time they will see that Diamond/Challenger border, would be when they play with their 5's team the next season.