Contents

# Line|Definition & Meaning

## Definition

A geometrical one-dimensional object that joins two or more points through a straight path is called a line. A line has no thickness and has no endpoints. A line with endpoints is referred as a line segment.

**y = mx + b** is the equation of a line of the** slope-intercept** form**,** **m** being the slope of the line and **b** being the y-intercept.

A** slope of a line** is the **ratio **where the** ch****ange** in the** y-coordinate **is the numerator and the change in the** x-coordinate** is the denominator, between two points on a line. Two points on a line can define it and can be represented by the linear equation that passes through those two points.

In addition to its use in **coordinate geometry,** lines are also used in other areas of mathematics, such as **calculus,** where they are used to represent the **graph of a function**, and in** geometric constructions**, where they are used to c**reate shapes and figures.**

## Drawing a Line

To draw a line in mathematics, you can use different forms of equations to describe it. Two common forms are the** slope-intercept form (y = mx + b)** and the** point-slope form (y – y1 = m(x – x1)).**

### Slope-intercept Form

In this form, the line is defined by its **slope (m)** and** y-intercept (b)**. To draw the line, you can start by** plotting the y-intercept** on the y-axis and then use the slope to determine the** direction and steepness of the line.**

### Point-slope Form

In this form, the line is defined by a **point on the line (x1, y1)** and its** slope (m)**. To draw the line, you can start by plotting the point on the** coordinate plane** and then using the **slope** to determine the **direction and steepness of the line.**

In both cases, once you have the **equation of the line,** you can **plot the points** that satisfy the **equation** and connect them with a **straight line.**

Overall, lines are a fundamental concept in mathematics that is used to represent geometric objects and to solve mathematical problems involving linear equations and functions.

## Types of Lines

There are several types of lines that are used in different contexts. Some of the most common types of lines include:

### Line Segment

A line segment is a piece of the line that has** two distinct endpoints.** It is a** limited portion** of a line, and it is defined by its two endpoints. A line segment can be represented graphically in a** coordinate plane** by plotting its two **endpoints** and **connecting** them with a **straight line**.

The distance between the two endpoints is known as the **length** of the line segment and can be calculated using the **distance formula.**

### Straight Lines

A straight line is a line that** extends infinitely** in both directions and has **no curvature**. A straight line is usually represented by the equation** y = mx + b**, **m being the slope** of the line and **b being the y-intercept.**

### Parallel Lines

Parallel lines are two or more lines that will always be at the** same distance** from each other and** never meet**, no matter their length.

- They are always at the same distance.
- They do not intersect.
- They have the same slope.
- They are always coplanar (in the same plane).
- The alternate interior angles of a transversal line crossing two parallel lines are congruent.

### Perpendicular Lines

Perpendicular lines are the two lines that intersect and form a **“****+” **because they intersect at 90 degrees.

- The angle between two perpendicular lines is always a right angle at 90 degrees.
- They intersect at exactly one point.
- They form four right angles when they intersect.

### Intersecting Lines

Intersecting lines can be **two or more lines** that will **cross** each other at **one or more points.** They intersect at least one point.

- The angles formed where intersecting lines meet are called vertical angles and are congruent.
- They have different slopes.
- They form four angles when they intersect.

### Horizontal Lines

A horizontal line is flat and goes **left to right along the x-axis** and has a constant** y-coordinate.** It has a** slope of zero**, and its equation is** y = b**, **b **being the **y-coordinate **of the line**.**

- They have a slope of zero.
- They are parallel to the x-axis.
- They have a constant y-coordinate.
- They are always equidistant from the x-axis (which is a line itself).

### Vertical Lines

A vertical line is a line that runs from** up to down along the y-axis** and has a **constant x-coordinate**. It has an** undefined slope**, and its equation can be written in the form** x = a, **where** a is the x-coordinate of the line.**

- They have an undefined slope.
- They are parallel to the y-axis.
- They have a constant x-coordinate.
- They are always equidistant from the y-axis (which is a line itself).

### Three Dimensional Lines

In three-dimensional space, lines are known as** “lines in space” or “space lines.”** They can be represented using different forms, such as **parametric form, vector form, or symmetric form.**

#### Parametric Form

A space line can be represented by a set of parametric equations, x = x1 + at, y = y1 + bt, z = z1 + ct, where (x1, y1, z1) being points on the line, (a, b, c) being the direction vector of the line, and t is a real parameter.

#### Vector Form

A point on the line can represent a space line and a direction vector, L: (x, y, z) = (x1, y1, z1) + t(a, b, c), where (x1, y1, z1) is a point on the line, (a, b, c) is the direction vector, and t is a real parameter.

#### Symmetric Form

A space line can be represented by the intersection of two planes, L: ax + by + cz + d = 0, where (a, b, c) is a normal vector of the line and d is a constant.

In three dimensions, lines can also intersect, be** parallel or skew lines.** A skew line is a line that is not parallel and not intersecting.

## Example Question

What will be the equation of the line with a y-intercept of 5 and a slope of -2?

### Solution

The slope-intercept form of the equation of a line is y = mx + b, where m and b are the slope and y-intercept, respectively.

Given the slope of -2 and the y-intercept of 5, we can write the equation of the line as:

y = -2x + 5

We can verify this by plugging in a point that lies on the line, for example (1,1):

y = -2(1) + 5 = 3

This equation represents a line that has a slope of -2 and a y-intercept of 5, which means that it has a negative slope (going down) and passes through the point (0,5).

*All mathematical drawings and images were created with GeoGebra.*