Question 1086039
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This problem is for Algebra-I, not for Calculus.
So I will give here another (more adequate) solution.


<pre>
To get the common points, substitute y = 6x + b into the first equation.  You will get

6x + b = {{{x^2 + 2x + 7}}}.


It is the same as

{{{x^2 - 4x + 7-b}}} = 0.   (1)


The original system has a unique solution if and only if the equation (1) has a unique solution.

It happens if and only if the discriminant of the equation (1) is zero:

d = {{{4^2 - 4*(7-b)}}} = 0,   or

16 = 4*(7-b),   or  4 = 7-b,   or   b = 7-4 = 3.


<U>Answer</U>.  b = 3.
</pre>

Solved.