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 « Basic Concepts
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 (0,b) 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:

### Two Point Form

If two points are available we will use the two point form equation for a line,

The slope formula is

Example 4:

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

Solution:

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

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

### 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