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Analytic Geometry: (lesson 2 of 4)

Lines

Only two pieces of information are needed to completely describe a given line. However, we have some flexibility on which two pieces of information we use

1. specifying the slope and the "y intercept", b, of the line (slope - intercept form).

2. Specifying the slope of the line and one point on the line (point slope form).

3. Specifying two points through which the line passes (two point form).

Slope-Intercept Form

The most useful form of straight-line equations is the "slope-intercept" form:

y = mx + b

This is called the slope-intercept form because "m" is the slope and "b" gives the y-intercept. (That means the point (b,0) is where the line cross the y-axis.)

The Slope-Intercept Form of the equation of a straight line introduces a new concept, that of the y-intercept. The y-intercept describes the point where the line crosses the y-axis. (At this set of coordinates, the 'y' value is zero, and the 'x' value is the y-intercept.)

Example 1:

1. y = 5x + 7

2. y = -3x + 23

3. y = 2x (or y = 2x + 0)

Point-Slope Form

The other format for straight-line equations is called the "point-slope" form. Suppose that we want to find the equation of a straight line that passes through a known point and has a known slope. For this one, they give you a point (x1, y1) and a slope m, and have you plug it into this formula:

y - y1 = m(x - x1)

Example 2:

1. y - 4 = -2(x - 1)

2. y - 8 = 3(x - 2)

3. y - 12 = 4(x - 3)

Example 3:

Find the equation of a line passing through the point (4, 2) and having a slope of 3.

Solution:

Slope-Intercept Form

Two Point Form

If two points Two Point Formare available we will use the two point form equation for a line,

Two Point Form

The slope formula is Two Point Form

Example 4:

Find the equation of the line that passes through the points (2, 4) and (1, 2).

Solution:

lines example

Example 5:

Find the equation of a line through the points (1,2) and (3,1). What is its slope? What is its y intercept? Solution: We first find the slope of the line by finding the ratio of the change in y over the change is x. Thus

Two Point Form equation

Now we can use the point - slope form to obtain:

lines example

lines example

Standard Form

In the Standard Form of the equation of a straight line, the equation is expressed as:

Ax + By = C

where A and B are not both equal to zero.

1. 7x + 4y = 6

2. 2x - 2y = -2

3. -4x + 17y = -432