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  Straight Lines Homework Help - K-12 Grade Level, College Level Mathematics

Introduction of Straight Line

The meaning of "Straight line" described here and it may be compared with the meaning of "circle" as keeping of those points in a plane that are a given distance (the radius) away from a mentioned point, that is called as center point . Circle is a some set of points selected according to their relation to some given parameters (center & radius). Likewise, straight line is a set of points selected according to their relation to some given objects (points & numbers). though, a circle determines its center and radius uniquely; for a straight line the situation is dissimilar from circle.

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Definition via distances

Let two different points X and Y be given. The set of all points Z such that

   ¦XY¦2 +¦XZ¦2=¦YZ¦2

is a straight line.

This is the straight line orthogonal to the line XY through the point X.

The condition ¦XY¦2 +¦XZ¦2=¦YZ¦2  means that the straight lines XY and XZ are orthogonal, that is, the angle YXZ is right (unless Z=X) .

Some properties of lines

Below, all points and lines are positioned in the plane (assumed to be a two-dimensional Euclidean space).

Most basic properties

For each two different points there exists one and keeps only one straight line that contains these two points.

Every straight line contains at least 2 different points.

There exist three points then there not lying on a straight line.

For each and every straight line and every point external the line there exists one and only one straight line through the mentioned point that does not intersect the given straight line.

Further properties

Two straight lines either does not intersect (are parallel), or intersect in a single point, or coincide.

Two straight lines at right angles or perpendicular to same line are parallel to each other (or coincide).

General (or standard) form

General form of the linear equation:

Ax +By = C

Where A, B ≠ 0

Can also be written as: ax + by + c = 0

Where A, B ≠ 0

Slope-intercept form

y = mx + b

Where m - slope of the line

b - y-intercept,

Point-slope form y - y1 = m(x - x1)

Where m - slope of the line

(x1,y1) - any point on the line.

Two-point form

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Where (x1, y1) and (x2, y2) are two points on the line with x2 ≠ x1.

Symmetric form: Multiplying both sides of this equation by (x2 - x1)

(x2 - x1)(y - y1) = (y2 - y1)(x - x1)

Intercept form

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Where a, b ≠ 0

Matrix form

Ax+ By = C

One can alter the equation in matrix form:

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Additionally, this representation extends to systems of linear equations.

A1x + B1y = C1

A2x + B2y = C2

Becomes

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