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.
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
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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, these lines are perpendicular.
