LINES

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WHAT IS A LINE?• From geometry, we already know that a line is determined

by two distinct points; and that it is composed of infinitely many points.

• Since a line is a set of points, technically, it is also a locus.

• A line is a locus of points that have a constant slope.

• A line intersects the x and y axes at the points which are called the x-intercept and y-intercept, respectively.

PROPERTIES OF A LINE

• constant slope (m)

• infinitely many points. • x - intercept

• y - intercept

y - intercept

x - intercept

constant slope

constant slope

constant slope

y - intercept

x - intercept

SLOPE OF A LINE• The slope of a line is determined by the ratio of the difference of the y-coordinates to the x-coordinates of two points on the line.

• It is represented as m and describes the steepness of the line it represents.

m = 0m > 0 m < 0

m = undefined

• also tells if lines are parallel, perpendicular, or simply intersects.

slopes are equal slopes are negative reciprocals

slopes are not equal nor negative reciprocals

• An intercept of a line is the point that the line intersects with the x or y axis.

• The point on the x-axis that a line intersects is the x-intercept while the point on the y-axis that it intersects is the y-intercept.

• An x-intercept is written as (a, 0) while a y-intercept is written as (0, b), for any real number a and b.

• A vertical line does not have a y-intercept, while a horizontal line does not have an x-intercept.

X AND Y INTERCEPTS

• A line is represented by an equation of the form Ax + By + C = 0, where A, B, and C are integers.

• The properties of a line such as the slope, intercepts, or any pair of points can create the equation of a line.

• There are four main forms of the equation of a line, all of which are based on the mentioned properties.

EQUATION OF A LINE

Two-point Form

Point-slope Form

Slope-intercept

FormIntercepts Form

• This form is used when two points of a line are given. These points are represented by (x1, y1) and (x2, y2).

Example:

Find the equation of the line that passes through the points and .

TWO-POINT FORM

**Should be written in Ax + By + C = 0 form.

• This form is used when the slope and a point of a line are given. The point is represented by (x1, y1) and the slope is m.

Example:

Find the equation of the line that passes through the points m = and .

POINT-SLOPE FORM

• This form is used when the both the x and y intercepts of a line are given. The x-intercept is represented by (a, 0) and y-intercept is represented by (0, b).

Example:

Find the equation of the line that passes through the points and .

INTERCEPTS FORM

NOTE: Two-point form can be applied if given the two intercepts.

• This form is used when the both the y intercept and the slope of a line are given. The slope is represented by m and y-intercept is represented by (0, b).

Example:

Find the equation of the line that passes through the points m = and .

SLOPE - INTERCEPT FORM

• The slope-intercept of a line is actually helpful even if the y-intercept is not given.

Example:

Find the equation of the line that passes through the points m = and .

USING THE SLOPE-INTERCEPT FORM GIVEN SLOPE BUT NOT Y-INTERCEPT

We don’t have b, but we know values

for x and y.

Solve for b, then

substitute it back to

the equation.

• The properties of a line that create the equation also allows us to graph the actual line.

• GIVEN TWO POINTS …

Example: and

GRAPH OF A LINE

• Just plot the points.

• Connect the points with a straight line.

• This is also the approach when the x and y intercepts are given.

• GIVEN A POINT AND A SLOPE…

Example: and m =

GRAPH OF A LINE

• Just plot the point.

• Get a second point by using the slope.

(in this example, the slope tells us that another point is two units down and three units to the right of the given point)

•Connect the points with a straight line.

• This is also done when the slope and an x or y intercept is given.

• GIVEN THE EQUATION

Example:

GRAPH OF A LINE

• Express the equation in slope-intercept form.

(for this example, it is )

• Use the y-intercept and the slope the same way the previous example was done.

• Connect the points with a straight line.

SOME NOT-SO-BASIC SITUATIONS INVOLVING LINES

• Remember how slopes of parallel and perpendicular lines are related. This is used in determining equations of lines through a point given a line parallel or perpendicular to it.

PARALLEL AND PERPENDICULAR LINES

slopes are equal slopes are negative reciprocals

What is the equation of the line that passes through (-3, 5) and parallel to the line given by ?

SOME NOT-SO-BASIC SITUATIONS INVOLVING LINES

The slope of the given line will be the

slope of the line being asked for

because they are parallel.

Since we have a slope and a point for the line being asked for, we can use point-slope

form.

What is the equation of the line that passes through (0, -9) and perpendicular to the line given by ?

SOME NOT-SO-BASIC SITUATIONS INVOLVING LINES

The negative reciprocal of the slope of the given line will be the slope of the line being asked for because they are

perpendicular.

Since we have a slope and a y-

intercept for the line being asked for, we can use slope-intercept

form.

LINE FAMILIES