# 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:
$d=\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}}$

#### 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.