Point Slope Equation of Line - X and Y Intercepts

Point

With a view to writing if you touch a piece of paper with a pen or pencil, a point is produced.

Another way ...

By intersecting two lines a points is produced. That is, the position at which two lines intersect is called a point. Two edges of a book, for example, meet at a point.

A point has no length, width or depth.

It is just a position.

Thus a point is of no dimension or zero dimension.

Line

A line is the set of points that extends along a straight path infinitely on and on its opposite direction.

Another way ...

A line is a continuous extent of straight length infinitely far on and on its opposite direction without width or depth.

So a line, strictly speaking, is perfectly straight and has no ends in both directions.

It has only length.

So, a line is of one dimension and sometimes called one d.

A line is shown in the figure that extends infinitely in both directions.

Suppose A and B are two distinct points on the line.

It is read AB is a line and denoted by

Again, one can have a concept of line from plane. For instance, if we decrease the width of a page gradually and make it at last to zero (0) keeping the length of the page fixed, it produces a line. This concept of line is derived from plane.

Thus a line in geometry includes the following two ideas:

• Line segment
• Ray

Ray

A ray is a part of a line starts from an end point and goes straight infinitely. Thus it has an end point.

To dive deep into line segment and ray in geometry, you can have a look at the complete tutorials on line segment and ray in other two different tutorials.

Slope of a Line

Let P(-3,8) and Q(-5,11) be two points of a straight line, 3x + 2y = 7

So, slope of the line PQ;

m = y₂ - y₁x₂ - x₁

বা, m = 11-8-5 - (-3)

বা, m = 3-5 + 3

বা, m = 3-2

∴ m = - 32

Equation of a line

One can determine the equation of a line depending on the given conditions. Equation of a straight line can be formed depending on points, slope and intercept of axes. Generally, equation of line can be determined depending on the minimum conditions as follows:

• Point slope form equation
• Slope y intercept form
• Slope x intercept form
• Equation of a line given two points
• Intercept form

Derivation of equation formula depending on given slope and a point of a line

Second Method

Again, Slope of the line passing through the points (x₁,y₁) and (x₂,y₂)

m = y₂ - y₁x₂ - x₁

where x₁ ≠ x₂; and the equation

y - y₁y₂ - y₁ = x - x₁x₂ - x₁

or, y - y₁x - x₁ = y₂ - y₁x₂ - x₁

or, y - y₁x - x₁ = m

∴ y - y₁ = m(x - x₁)

This is the required equation of straight line.

Derivation of equation formula depending on given slope and intercept of any axis.

Equation of a straight line can be determined on given slope and the length of the part of x-axis or y-axis cut by the line.

What is intercept?

The part of axes cut by the straight line is called the intercept of relevant axis. More precisely, the point on any axis at which the straight line intersect, the distance of that point from the origin (0,0) is called the intercept of the relevant axis. For instance, if a straight line intersect the x-axis at a point (b,0), the intercept of x-axis is b. Similarly, if a line intersect the y-axis at a point (0,c), the intercept of y-axis is c.

Equation of line can be determined in two different ways depending on given slope and intercept of axes.

• Slope and y intercept form
• Slope and x intercept form

Finding the equation of line depending on given slope and y intercept

Putting the value of y₁ in original equation, we get

y - y₁= m(x - x₁)

or, y - (c + mx₁) = m(x - x₁)

or, y - c - mx₁ = mx - mx₁

or, y = mx - mx₁ + c + mx₁

∴ y = mx + c

which is the required equation.

Deriving the equation of line depending on given slope and x intercept

Let the slope of the equation be m and the intercept of x-axis be b, that is the line intersect the x-axis at (b,0).

We have already learned to find the equation of a straight line with slope m and passing through the point (x₁, y₁) is

y - y₁ = m(x - x₁)

The line above goes through the point (b,0), so we get

0 - y₁ = m(b - x₁)

or, 0- y₁ = m(b - x₁)

or, - y₁ = bm - mx₁

or, - y₁ = -(-bm + mx₁)

∴ y₁ = mx₁ - bm

Putting the value of y₁ in original equation,

y - y₁ = m(x - x₁)

or, y - (mx₁ - bm) = m(x - x₁)

or, y - mx₁ + bm = mx - mx₁

or, y = mx - mx₁ + mx₁ - bm

or, y = mx - bm

or, y = m(x - b)

or, m(x - b) = y

or, x - b = 1m y

or, x = 1m y + b

or, x = ny + b

where n = 1m that is, mn=1.

which is the required equation.

Equation of a straight line on given two points

One can find the equation of a line passing through two points. Let (x₁, y₁) and (x₂, y₂) be two points of a straight line.

Again, let the slope of the line be m. It is described in this tutorial the equation of a line whose slope m and passing through the point (x₁, y₁) is

y - y₁ = m(x - x₁)

This line passes through the point (x₂, y₂). So, we get

y₂ - y₁ = m(x₂ - x₁)

or, m(x₂ - x₁) = y₂ - y₁

∴ m = y₂ - y₁x₂ - x₁

Putting the value of m in the original equation, we get

y - y₁ = m(x - x₁)

or, y - y₁ = y₂ - y₁x₂ - x₁ (x - x₁)

∴ y - y₁y₂ - y₁ = x - x₁x₂ - x₁

which is the required equation of the straight line.

Equation of a line depending on x and y intercepts

One can find the equation of a straight line if the intercept of x-axis and intercept of y-axis are given.

Let the intercept of x-axis and y-axis be a and b respectively. That is, the line intersect the x-axis at P(a,0) and intersect the y-axis at Q(0,b) where a\neq 0,b\neq 0; and slope of the line is m.

The line above passes through the point (0,b). So, we get

b = m(0 - a)

or, b = m(- a)

or, b = -ma

or, -ma = b

∴ m = - ba

Now, write the main equation and then put the value of m.

y = m(x - a)

or, y = - ba (x - a)

or, y = - bxa + b

or, bxa + y = b

Dividing both sides by b,

or, bxab + yb = 1

∴ xa + yb = 1

This is the standard equation of straight line.