# Introduction to Straight Lines

### Introduction to Straight Lines in Two-Dimensional Coordinate Geometry

#### Cartesian Coordinate System:

- Two-dimensional space is represented using a Cartesian coordinate system, consisting of x-axis and y-axis intersecting at the origin (0,0).
- Points on this plane are represented by ordered pairs (x, y), where 'x' denotes the horizontal distance and 'y' denotes the vertical distance from the origin.

#### Equation of a Straight Line (Linear Equation):

- A straight line in a Cartesian plane can be defined by the equation
**y = mx + c**or**Ax + By + C = 0**, where:- 'm' represents the slope of the line (rate of change of y with respect to x).
- 'c' represents the y-intercept (where the line crosses the y-axis).
- 'A', 'B', and 'C' are coefficients in the general form of the equation.

#### Slope-Intercept Form:

- The slope-intercept form of the equation is
**y = mx + c**, where 'm' is the slope and 'c' is the y-intercept. - The slope 'm' indicates the direction and steepness of the line:
- Positive slope: Line slants upwards from left to right.
- Negative slope: Line slants downwards from left to right.
- Zero slope: Line is horizontal.
- Undefined slope: Line is vertical.

#### Point-Slope Form:

- The point-slope form of the equation is
**y - y₁ = m(x - x₁)**, where 'm' is the slope and (x₁, y₁) is a point on the line. - This form allows for easily deriving the equation when the slope and a point on the line are known.

#### Parallel and Perpendicular Lines:

- Parallel lines have the same slope but different y-intercepts.
- Perpendicular lines have slopes that are negative reciprocals of each other (the product of their slopes is -1).

#### Intercepts:

**x-intercept**: The point where the line crosses the x-axis (y = 0).**y-intercept**: The point where the line crosses the y-axis (x = 0).

#### Distance between Points:

- The distance 'd' between two points (x₁, y₁) and (x₂, y₂) is given by the formula:
$$d=\sqrt{({x}_{2}-{x}_{1}{)}^{2}+({y}_{2}-{y}_{1}{)}^{2}}$$

#### Angle between Lines:

- The angle 'θ' between two lines with slopes 'm₁' and 'm₂' is given by the formula:
$$\mathrm{tan}\theta =\frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}$$

#### Applications:

- Straight lines find applications in various fields such as engineering, physics, computer graphics, and architecture.
- They are used to model and analyze many real-world phenomena, including motion, electricity, and geometry of shapes.

#### Summary:

- Equations of straight lines are foundational in coordinate geometry, describing the relationship between points and their graphical representation on a Cartesian plane.
- Different forms of equations allow for diverse methods to describe and analyze straight lines, facilitating problem-solving in various contexts.