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 yintercepts. 
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 pointslope form, y  y_{1}
= m( x  x_{1}), use the point ( x_{1}, y_{1})
and the slope m. When the equation is in slopeintercept
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. 

Slopeintercept form. 
yintercept: 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 slopeintercept 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 ( x_{1}, y_{1})
and ( x_{2}, y_{2}) 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) = ( x_{1}, y_{1}).
Form two equations by setting the xcoordinates equal to each
other and the ycoordinates equal to each other.
Coordinates of the other endpoint: (5, 1).
