Unit 2: The Straight Line

Various Forms of Equation of a Straight Line

An equation of a straight line is an algebraic relationship between x and y that is true for every point on the line and false for every point not on the line.

General Form

The most general form of a linear equation.

Ax + By + C = 0

From this, the slope is m = -A/B and the y-intercept is -C/B.

Slope-Intercept Form

This is the most common form, useful for graphing.

y = mx + c

• m = slope of the line (change in y / change in x).
• c = y-intercept (the point where the line crosses the y-axis, (0, c)).

Point-Slope Form

Used when you know the slope m and one point (x₁, y₁) on the line.

y - y₁ = m(x - x₁)

Two-Point Form

Used when you know two points (x₁, y₁) and (x₂, y₂) on the line.

First, find the slope: m = (y₂ - y₁) / (x₂ - x₁).
Then, substitute this into the Point-Slope form:

y - y₁ = [ (y₂ - y₁) / (x₂ - x₁) ] · (x - x₁)

Intercept Form

Used when you know the x-intercept a (point (a, 0)) and the y-intercept b (point (0, b)).

x/a + y/b = 1

Normal (or Perpendicular) Form

This form is defined by the length of the perpendicular (the "normal") from the origin to the line, and the angle this normal makes with the positive x-axis.

x cos(α) + y sin(α) = p

• p = length of the normal from the origin (must be positive).
• α = angle the normal makes with the positive x-axis.

Angles Between Two Straight Lines

Formula for Angle

Let two lines have slopes m₁ and m₂. If θ is the acute angle between them:

tan(θ) = | (m₁ - m₂) / (1 + m₁m₂) |

If the lines are in general form, A₁x + B₁y + C₁ = 0 and A₂x + B₂y + C₂ = 0, then m₁ = -A₁/B₁ and m₂ = -A₂/B₂. Substituting these gives:

tan(θ) = | (A₂B₁ - A₁B₂) / (A₁A₂ + B₁B₂) |

Condition for Parallel Lines

Parallel lines have the same slope. The angle θ between them is 0°, and tan(0) = 0. This means the numerator in the formula must be zero.

Condition: m₁ = m₂
(For general form: A₁/A₂ = B₁/B₂ or A₁B₂ - A₂B₁ = 0)

Condition for Perpendicular Lines

Perpendicular lines have slopes that are negative reciprocals. The angle θ is 90°, and tan(90) is undefined. This means the denominator in the formula must be zero.

Condition: m₁m₂ = -1
(For general form: A₁A₂ + B₁B₂ = 0)

Length of Perpendicular

This is the shortest distance from a given point P(x₁, y₁) to a given line Ax + By + C = 0.

[Image of the distance from a point to a line]

Distance d = | (Ax₁ + By₁ + C) / sqrt(A² + B²) |

Intersection of Two Straight Lines

The point of intersection is the single point (x, y) that satisfies *both* line equations simultaneously.

Given two lines:
A₁x + B₁y + C₁ = 0
A₂x + B₂y + C₂ = 0

To find the intersection, you must solve this system of two linear equations. You can use any of these methods:

1. Substitution: Solve one equation for y (e.g., y = ...) and substitute that expression into the second equation.
2. Elimination: Multiply the equations by constants so that the coefficients of either x or y are opposites, then add the equations together.
3. Cramer's Rule (using determinants):
   x = (B₁C₂ - B₂C₁) / (A₁B₂ - A₂B₁)
y = (C₁A₂ - C₂A₁) / (A₁B₂ - A₂B₁)

Unit 2: Exam Quick Tips