Answer
$rate=k[NH_{3}][BF_{3}]$
Work Step by Step
From Experiment #1 to Experiment #2, $[BF_{3}]$ is held unchanged and $[NH_{3}]$ is reduced by half ($0.250/0.125=2$). After the change, the proportion between the initial rates of Experiment #1 and #2 is: $0.2130/0.1065=2$. Because $2^1=2$, the rate of reaction is first-order in $BF_{3}$.
Similarly, from Experiment #4 to Experiment #5, $[NH_{3}]$ is held unchanged and $[BF_{3}]$ is reduced by half ($0.350/0.175=2$). After the change, the proportion between the initial rates of Experiment #1 and #2 is: $0.1193/0.0596\approx2$. Because $2^1=2$, the rate of reaction is also first-order in $NH_{3}$.
Thus, we are able to construct the rate law for this reaction:
$rate=k[NH_{3}][BF_{3}]$
