# Slope of the Line Joining Two Points

We will discuss here about the slope of the line joining two points.

To find the slope of a non-vertical straight line passing through two given fixed points:

Let P (x$$_{1}$$, y$$_{1}$$) and Q (x$$_{2}$$, y$$_{2}$$) be the two given points. According to the problem, the straight line PQ is non-vertical x$$_{2}$$ ≠ x$$_{1}$$.

Required to find, the slope of the line through P and Q.

From P, Q draw perpendiculars PM, QN on x-axis and PL ⊥ NQ. Let θ be the inclination of the line PQ, then ∠LPQ = θ.

From the above diagram, we have

PL = MN = ON - OM = x$$_{2}$$ - x$$_{1}$$ and

LQ = = NQ - NL = NQ - MP = y$$_{2}$$ - y$$_{1}$$

Therefore, the slope of the line PQ = tan θ

= $$\frac{LQ}{PL}$$

= $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$

= $$\frac{Difference of ordinates of the given points}{Difference of their abscissae}$$

Hence, the slope (m) of a non-vertical line passing through the points P (x$$_{1}$$, y$$_{1}$$) and Q (x$$_{2}$$, y$$_{2}$$) is given by

slope = m = $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$

1. Find the slope of the line passing through the points M (-2, 3) and N (2, 7).

Solution:

Let M (-2, 3) = (x$$_{1}$$, y$$_{1}$$) and N (2, 7) = (x$$_{2}$$, y$$_{2}$$)

We know that the slope of a straight line passing through two points (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) is

m = $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$

Therefore, slope of MN = $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$ = $$\frac{7 - 3}{2 + 2}$$ = $$\frac{4}{4}$$ = 1.

2. Find the slope of the line passing through the pairs of points (-4, 0) and origin.

Solution:

We know that the coordinate of the origin is (0, 0)

Let P (-4, 0) = (x$$_{1}$$, y$$_{1}$$) and O (0, 0) = (x$$_{2}$$, y$$_{2}$$)

We know that the slope of a straight line passing through two points (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) is

m = $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$

Therefore, slope of PO = $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$

= $$\frac{0 - (0}{0 - (- 4)}$$

= $$\frac{0}{4}$$

= 0.

Equation of a Straight Line