# 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)

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P(-6, 2b + 3) , Q(7, -1)

Parallel to the x axis means that all x values have the same y value for your line.

so 2b + 3 = -1

2b = -1 - 3 = -4

b = -4/2 = -2

P(-6, 2b + 3) , Q(7, -1)

if you want to make P and Q be parallel to the x axis

you just need to make sure that 2b+3=-1, that means P and Q have the same distance from x-axis, so P and Q will be parallel to the x axis

from 2b+3=-1,you can get that b=-2

i think this should be the easiest way

A line parallel to x-axis make an angle 0 with x-axis.

Therefore when the line PQ is parallel to x-axis it will make and angle 0 with x-axis. Therefore, the slope of PQ will be:

= tan0 = 0

The coordinates of P (x1, y1), and Q (x2, y2) are given as:

x1 = -6

y1 = 2b + 3

x2 = 7

y2 = -1

Slope of PQ is given by the formula:

Slope = (y2 - y1)/(x2 - x1) = [-1 - (2b + 3)]/[7 -(-6)]

= [-1 - 2b - 3)]/(7 + 6)

= (- 2b - 4)/13

As explained above, when PQ is parallel to x-axis this slope is 0.

Therefore:(- 2b - 4)]/13 = 0

Therefore: - 2b - 4 = 0

Therefore: - 2b = 4

Therefore b = -4/(-2) = -2

Answer:

The line PQ will be parallel to x-axis when b = -2.

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