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Linear Equations

Definitions

Coordinate Plane: Two number lines drawn perpendicular to each other.

Quadrants: The four regions defined by the boundaries create by the number lines forming a coordinate plane.

X-Axis: The Horizontal number line in a coordinate plane.

Y-Axis: The Vertical number line in a coordinate plane.

Ordered Pair: The location of a point on a coordinate plane written as (x, y), the x-coordinate and the y-coordinate.

The Origin: The point where the two number lines meet in forming a coordinate plane having the location of the ordered pair (0, 0)

Examples

Quadrant IQuadrant II

Quadrant III Quadrant IV

A

B

C

D

Find the locations for points A, B, C, and D.

A(-2, 3) B(-3, -3)

C(3, -2) D(4, 5)

Linear Equations

Definitions

Slope: The ratio in change in the Y direction to the change in the X direction between two points on a line segment. Also known as, the ratio of rise over run.

scoordinate-in x changescoordinate-yin change

runrise

m ¸¹·

¨©§

¸¹·

¨©§

Formulas

Slope of a Line: given two

points on a line (x1, y1) and (x2, y2).

,)1x2(x)1y2(y

m��

Slope of a Horizontal Line: A horizontal line containing point (a, b) is described by the equation y = b and has a slope of 0.

Slope of a Vertical Line: A vertical line containing point (c, d) is described by the equation x = c and has an undefined slope.

Linear Equations

A

B

Example

Find the slope of .AB

Linear Equations

Exercises

A

B

C

D

E

FGH

Find the slopes of the following lines.

AB1) 2)

3) 4)

CD

EF GH

Linear Equations

Examples

Find the slope of the line that passes through points (1, -1) and (-3, -1).

Find the slope of the line that passes through points (-2, 3) and (4, 0).

Linear Equations

On Your Own

Find the slope of the line passing through the sets of points given below.

1) A(2, 3) and B(8, 6)

2) C(-4, -3) and D(-8, 6)

3) E(2, 2) and F(2, -1)

Linear Equations

Parallel and Perpendicular LinesDefinitions

Perpendicular: Lines intersecting to form a right or 90o angle.

Reciprocal: The reciprocal of a fraction is the fraction where the numerator and denominator are switched.

Parallel Lines:

If two lines are parallel, they have the same slope.

If two distinct lines have the same slope, they are parallel.

Perpendicular Lines:

If two lines are perpendicular, the product of their slopes is -1. The slopes are negative reciprocals of each other.

If the slopes of two lines are negative reciprocals of each other, the line are perpendicular.

Linear Equations

Parallel and Perpendicular LinesExamples

Linear Equations

1) Find the negative reciprocal of 4

2) Find the negative reciprocal of 78

Exercises

Linear Equations

Determine if the lines going through the given point are parallel, perpendicular or neither.

1) The line through points (-5, -3) and (5, 3) and the line through points (7, 9) and (-3, 3).

2) The line through points (6, 5) and (12, 3) and the line through points (0, 2) and (4, -2).

3) The line through points (5, 1) and (1, 9) and the line through points (8, 7) and (-3, 29)