LEARN THINGS THE EASY WAY

# Point Slope Equation of Line - X and Y Intercepts

By the end of this lesson, you will be able to

define point in plane geometry.

explain line definition in geometry.

interpret slope of a line with picture.

recognize x and y Intercepts of line.

find the equation of line.

relate slope and intercept to equation of a line.

identify equation of line in different forms.

## 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.

### Straight Line Example

Lines and their equations

Thus a line in geometry includes the following two ideas:

• Line segment
• Ray

### Line Segment

A line segment is a finite part of a line. Therefore, it has two end points. Including end points, all the points between its end points lie on it.

A line segment is shown in the figure.

### 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.

## Apps for drawing line based on two known points

Input Points
x1= y1= x2= y2=

Line hits the points on edges

x1= y1= x2= y2=

## Slope of a Line

Changes in y coordinates divided by changes in x coordinates of two points of a straight line is called the slope of the line. Thus, the slope of a straight line is a number. This number may be positive or negative. Therefore, a line can have positive slope or negative slope. Again, the slope can also be zero. Slope of a line is denoted by m.

Suppose, P(x1,y1) and Q(x2,y2) are two points where x1 ≠ x2.

Thus, slope

m = changes in y coordinateschanges in x coordinates

বা, m = y2 - y1x2 - x1

বা, m = ΔyΔx

Two points and the slope of the line is shown in the figure.

Slope of a straight line passing through the points P(x1,y1) and Q(x2,y2) where x1 ≠ x2;

m = y2 - y1x2 - x1

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

#### First Method

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

m = y2 - y1x2 - x1

where x1 ≠ x2;

Let P(x1,y1) be a fixed point and Q be a variable point where Q(x,y) that is, Q(x2,y2) = Q(x,y). This implies the slope,

m = y2 - y1x2 - x1

বা, m = y - y1x - x1

∴ y - y1 = m(x - x1)

This is the required equation of straight line.

where x ≠ x1; because if x = x1, straight line disappears and it will be just a point.

Two points on a line are shown in the figure.

#### 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.

Eqution of a line whose slope is m and passing through the point (x1, y1)

y - y1 = m(x - x1)

### 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

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

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)

This line above passes through the point (0,c). So,

c - y1 = m(0 - x1)

or, c - y1 = m(- x1)

or, c - y1 = -mx1

or, c + mx1 = y1

∴ y1 = c + mx1

Slope and y intercept of a line is shown in the figure.

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.

The equation of a straight line whose slope m and y intercept c is

y = mx + c

#### 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 whose slope is m and x intercept b is

x = ny + b

where n = 1m অর্থাৎ, mn = 1.

### 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 straight line passing through two points (x1, y1) and (x2, y2) is

y - y1y2 - y1 = x - x1x2 - x1

### 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.

It is known to us, the equation of line with slope m and goes through (x1, y1) is

y - y1 = m(x - x1)

So, the equation of the line whose slope m and passes through the point (a,0) is

y - y1 = m(x - x1)

or, y - 0 = m(x - a)

or, y = m(x - a)

A straight line and intercept of both axes are shown in the figure.

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.

The equation of a straight line whose intercept of x-axis and y-axis be a and b respectively where a≠ 0, b≠ 0 is

xa + yb = 1