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Linear Equations
Linear Equations
Graphing Linear Equations
| Graphing Equations Using Two Points |
Use the equation to find the coordinates of any two
points on the line. Draw the line representing the equation by
connecting them. The two points chosen can be the x- and y-intercepts. |
| Graphing Equations Using a Point and the Slope |
Graph one point and use the slope to find another point
by moving the distance of the change in y and then the distance of the
change in x from that point. When the equation is in point-slope form, y
- y1
= m( x - x1), use the point ( x1, y1)
and the slope m. When the equation is in slope-intercept form, y = mx +
b, use the point (0, b) and the slope m. |
Example
Graph -2x + 3y = 9 by using the slope and y -intercept.
Solution
| 3y = 2x + 9 |
Solve the equation for y. |
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Slope-intercept form. |
y-intercept: 3  |
(0, 3) is on the line. |
slope of line:  |
Move up 2 units, then right 3 units from that point. |
Parallel and Perpendicular Lines
| Parallel Lines |
Lines in the same plane that never intersect are called
parallel lines. If two nonvertical lines have the same slope, then they
are parallel. All vertical lines are parallel. |
| Perpendicular Lines |
Lines that intersect at right angles are called
perpendicular lines. If the product of the slopes of two lines is -1,
then the lines are perpendicular. The slopes of two perpendicular lines
are negative reciprocals of each other. In a plane, vertical lines and
horizontal lines are perpendicular. |
Example
Determine whether the graphs of 2 y = -3 x + 4 and 3 y = 2 x - 9 are
parallel, perpendicular, or neither.
Solution
Rewrite each line in slope-intercept form to identify its slope.
| 2 y = -3 x + |
3 y = 2 x - 9 |
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Since  , these lines are perpendicular.
Midpoint of a Line Segment
| Midpoint of a Line Segment |
The midpoint of a line segment is the point that is
halfway between the endpoints of the segment. The coordinates of the
midpoint of a line segment whose endpoints are at ( x1, y1)
and ( x2, y2) are given by  |
Example
The midpoint of a segment is M (2, 3) and one endpoint is B ( -1, 5). Find
the coordinates of the other endpoint.
Solution
Let M(2, 3) = (x, y) and B( -1, 5) = ( x1, y1).
Form two equations by setting the x-coordinates equal to each other and the
y-coordinates equal to each other.
Coordinates of the other endpoint: (5, 1).
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