Mullinax settled for a share of second with countryman Sean O'Hair on 273 with American Jimmy Walker fourth on 274 and two-time major victor Zach Johnson fifth, another stroke adrift.
The 30-year-old Texan, busy at home in Austin with the recent birth of the family's first child, parred the final seven holes for a 4-under 68 and a two-stroke victory over Trey Mullinax and Sean O'Hair.
Mullinax left himself a nine-foot birdie putt while Landry needed to two-putt from 50 feet for the victory but rolled his first putt eight feet past the cup.
"It took a lot of hard work to get to this point", Landry said.
Landry played at Arkansas after starring at Port Neches-Groves High School east of Houston.
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Looking remarkably assured in his first pay-for-play appearance, Niemann used his textbook swing and confident putting stroke to card a five-under-par 67 at TPC Antonio. Landry, interviewed on camera after he had finished 21st on the 2015 Web.com Tour money list, was asked which players he looked up to.
The 30-year-old began the day in a tie at the top with Zach Johnson, but quickly assumed the outright lead with birdies on each of his three opening holes.
"Everybody was out", Landry said.
However, a bogey five at the 11th coupled with a charge which saw Trey Mullinax - who had posted a course record 62 on Saturday - reduce his lead to a single shot with three birdies in four holes around the turn, set up a potentially dramatic conclusion.
"I hope it does", he said. "I think that that was really, really key to getting to where I am right now". "Felt like if I could just hit a few more greens, because this golf course is all about that as well, make a few putts, which is what I did, I was going to have a good week".
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UN Ambassador Nikki Haley welcomed the announcement and praised the UN Security Council for coming together to achieve this goal. Japanese Prime Minister Shinzo Abe also welcomed the North Korean statement but said it must lead to action.
However, it was a particularly impressive outing for former World No.1 Amateur Joaquin Niemann, who finished in 6th place thanks to a final round 67 in his first PGA Tour event as a professional.
He is the fourth Texas-born player to earn his first career win in this event, joining Ben Crenshaw, Bart Bryant and Bob Estes. He earned his tour card previous year on the Web.com Tour, and lost a playoff to Jon Rahm in January in the CareerBuilder Challenge.
Walker ended the week in outright fourth on 14 under and one shot ahead of two-time victor Zach Johnson, who ran up a scrappy six at the last to complete a disappointing 72.
"This is an wonderful feeling, to be competing with the best players in the world and seeing my name on the leaderboard", Niemann said. "Now I just have to keep it going".
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That included positioning vehicles similar to those involved in the collision, taking measurements and analyzing sightlines. Thursday, two of them in critical condition, the Saskatchewan Health Authority said in an update on patient condition.