# Line and Definition of Slope

Slope of a line (steepness)

Consider a particle moving along a non vertical line segment from a point p_{1}( x_{1},y_{1} ) to a point p_{1}( x_{1},y_{1} ). The vertical change y_{2} – y_{1} is called the rise, and the horizontal change x_{2} – x_{1} the run.

DEFINITION

If **P(x _{1}, y_{1})** and

**P(x**are points on a nonvertical line, then slope

_{2}, y_{2})**m**of the line is defined by:

It does not matter which point is called P

_{1}and which one is called P

_{2}

**Slope of P**

_{1}P_{2}= (y

_{2}- y

_{1})/(x

_{2}- x

_{1})

= -(y

_{1}- y

_{2})/[-(x

_{1}- x

_{2})]

= (y

_{1}- y

_{2})/(x

_{1}- x

_{2}) = Slope of P

_{1}P

_{2}

Any two distinct points on a nonvertical line can be used to calculate the slope of the line. To measure the slope, we generally move from left to right when measuring the distance travelling **horizontally**.

Because of this, sometimes the concept of *fall* replaces thar of *rise*!

Example

In each part find the slope of the line through

**
(A) (6, 2) and (9, 8)
(B) (2, 9) and (4, 3)
(C) (-2, 7) and (5, 7)
**

Solution:

We know that slope of line through two points P_{1}(x_{1}, y_{1}) and p_{1}(x_{1}, y_{1}) , is given by

m= (y_{2} - y_{1})/ (x_{2} - x_{1})

So

a) m= (8 - 2)/(9 - 6) = 6/3 = 2

On coordinate plane *xy*

Similarly

b) m= (3 - 9)/(4 - 2) = -6/2 = -3

On coordinate plane *xy*

Also

c) m= (7 -7)/[5 - (-2)] = 0/7 = 0

On coordinate plane *xy*

Definition (Angle of inclination)

For a line L is not parallet to x-axis, the angle of inclination is the smallest angle φ messured counterclockwise from the direction of the positive x-axis to L.

For a line parallel to the x-axis, we take φ = 0

As shown in the following figures.

If **m** is slope of line then,

**m = rise/run**

= Rate of change of *y* with respect to *x*

THEOREM

For a nonvertical line, the slope **m** and the angle of inclination φ are related by

**m = tan φ **

Example:

Find the angle of inclination for a line of slope **m = 1** and the angle of inclination for a line of slope **m = -1**

Solution:

If m=1 tan φ = 1, so that φ = π/4 = 45°

If m=-1 tan φ = -1 since, 0 < φ < π **φ = 3π/4 = 135°**

THEOREM
Let L_{1} and L_{2} be lines with slopes *m _{1}* and

*m*, respectively

_{2}(a) The lines are parallel if and only if

**m**

_{1}= m_{2}(b) The lines are prependicular if and only if

**m**

_{1}m_{2}= -1
Proof: (a)

If L_{1} and L_{2} are non vertical lines, then their angles of inclination φ_{1} and φ_{2} are equal.

**φ _{1} =φ_{2} **

Thus

**m**

_{1}= tanφ_{1}= tanφ_{2}= m_{2}Conversely, if two slope lines are equal, I.e.

M

_{1}= M

_{2}

⇒ tan(φ

_{1}) = tan(φ

_{2})

⇒ φ

_{1}= φ

_{2}

So, lines are parallel.

(b) Assume that **φ _{1} < φ_{2}**

Then referring to the figure

**m**

m

_{1}= tanφ_{1}= c/hm

_{2}= tanφ_{2}= -h/cThe proof of the converse is left as an exercise.

THEOREM

The vertical line through (a, 0) and the horizontal line throudgh (0, b) are represented, respectively, by the equation

*x = a*and

*y = b*

THEOREM

The line Passing throudh P

_{1}(x

_{1}, y

_{1}) and having slope

*m*is given be the equation

y - y

_{1}= m(x - x

_{1})

This is called the

*point-slope*form of the line.

THEOREM

The line with y-intercept b and slope m is given by the equation

y = mx + b

This is called the slope-intercept form of the line.