jmc

If the curves $y_1$ and $y_2$ intersect at only one point, then there should be only one solution to the equation $x^2+2x+7=6x+b$. To find $b$, we first rearrange the equation to get $x^2-4x+(7-b)=0$. This equation has only one solution if and only if the discriminant of $x^2-4x+(7-b)=0$. Thus, we need $$\begin{array}{rl}16-4(7-b)& =0\Rightarrow \\ 4b& =12\Rightarrow \\ b& =\boxed{3}.\end{array}$$