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## 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. 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).