For what value of b will the line joining the points P and Q be parallel to the x axis? P(-6, 2b + 3) , Q(7, -1)

The slope of the  line joining the two points(x1,y1) and( x2,y2) is

(y2-y1)/(x2-x1).

So, the slope of the line joining P((-6,2b+3) and Q(7,-1) is (-1-(2b+3))/(7-(-6))=

(-4-2b)/13=-(4+2b)/13....................(1)

The slope of x axis is zero, as x axis is inclined  to itself at  angle zero and  so the slope of x axis is the tangent of zero angle  which is zero.

PQ and x axis are parallel only if the slope of PQ  equals  to that of x axis which is zero.This can be only  when  slope obtained of PQ in (1) ,that is, 4+2b/13  = 0 or b= -4/2 = -2.

So the value b=-2,makes PQ ||  to X axis.

Aliter:

Any line pararallel to the x axis keeps the same distance from  x axis  . So, the coordinates (or ordinates) of any point  on the line parallel to x axis should be same or equal.  That means, the value of the y coordinate is same for all points on the line paarallel to x axis.

Since PQ || ro X axis,

Y coordinate of P(-6,2b+1)  =  Y coordinate of Q(7 -3) or

2b+1= -3 or

2b=-3-1 or

b= -4/2 =-2