3-5: Vocabulary rise, run, slope point-slope form of a line slope-intercept form of a line equations for parallel and perpendicular lines The slope of a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope.

Example 1: Finding the Slope of a Line Use the slope formula to determine the slope of each line. AB AC CD

Opposite Reciprocals If a line has a slope of , the slope of a perpendicular line is The ratios and are called opposite reciprocals.

Example 2A: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or oblique. GH and IJ for G(–3, –2), H(1, 2), I(–2, 4), and J(2, –4)

Example 2B: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or oblique. CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3)

(4,3)

Examples 3A/B: Writing Equations In Lines Write the equation of: the line with slope 6 through (3, –4) in point-slope form. Convert to slope-intercept form. the line through (–1, 0) and (1, 2) in slope-intercept form

Example 3C/D/E: Writing Equations In Lines Write the equation of: the line with the x-intercept 3 and y-intercept –5 in point slope form the line with slope undefined through (4, 6) in slope-intercept form the line through (–3, 2) and (1, 2) in point-slope form

Example 4A: Graphing Lines Graph each line. The equation is given in the slope-intercept form, with a slope of and a y-intercept of 1. Plot the point (0, 1) and then rise 1 and run 2 to find another point. Draw the line containing the points. (0, 1) rise 1 run 2

Example 4B: Graphing Lines Graph each line. y = –3 The equation is given in the form of a horizontal line with a y-intercept of –3. The equation tells you that the y-coordinate of every point on the line is –3. Draw the horizontal line through (0, –3). (0, –3)

A system of two linear equations in two variables represents two lines A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms.

Examples 5A/B: Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. 2y – 4x = 16, y – 10 = 2(x - 1)

Writing Equations for Parallel or Perpendicular Lines to a Given Line Determine the slope of the given line. Use m║ or m and a point on the new line Example 6A: Given the line 2x + y = 1 Write the equation for a parallel line and a perpendicular line that pass through the point A(4, 8) (point-slope form). Then, put the new equation in slope-intercept form. Example 6B: Find the equation for the perpendicular bisector of AB with A(6, -4) B(10, 8) (pt-m form)

HONORS: Finding the Distance From a Given Point to a Given Line Determine the slope of the given line. Use m and the given point to write an equation of the line  to the given line Solve the system of equations to find the intersection point of the two lines Use the given point and the intersection point to calculate the distance Honors: Example 7A: Given the line 2x + y = 1 and the point A(4, 8) find the distance from the point to the line.  

HONORS: Example 7B: HONORS: Example 7C: Find the distance between the line 2x = y + 5 and the point R(6, 2)   HONORS: Example 7C: Find the distance between 2║lines y = 2x+4, y = 2x+10 (Use (0, 10) as one of the points for the distance)