Algebra 1 Section 6.5

Linear Equations Standard form: Ax + By = C Slope-intercept form: y = mx + b Point-slope form: y – y1 = m(x – x1)

Point-Slope Form of a Line The point-slope form provides an alternative method of determining the equation of a line from the slope (m) and any given point (x1, y1) on the line.

Example 1 Equation of the line passing through (-5, 2) and (3, -4): First, find the slope: m = y2 – y1 x2 – x1 -4 - 2 3 -(-5) = -6 8 = 3 4 = -

Example 1 y – y1 = m(x – x1) y -(-4) = -¾(x – 3) y + 4 = - x + 3 4 9 7

Example 1 y = - x – 3 4 7 To standard form: 4y = -3x – 7 3x + 4y = -7

Definitions Parallel lines are lines in the same plane that have no points in common. Perpendicular lines meet at right angles (90°).

Parallel Lines The graphs of linear equations are parallel if they have the same slope but different y- intercepts.

Example 2 3x – y = 6 y = 3x – 6 m1 = 3 y-int: (0, -6) -9x + 3y = 12 The lines are parallel.

Example 3 Find the slope of 4x + 3y = 17. 4 m = - 3 The parallel line has the same slope, and goes through the point (-5, 9).

Example 3 y – y1 = m(x – x1) y – 9 = - [x -(-5)] 4 3 y – 9 = - x – 4 3 20 y = - x + 4 3 7

Slopes Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other.

Number Negative Reciprocal Example 4 Number Negative Reciprocal 8 5 5 8 - 1 9 - 9

Number Negative Reciprocal Example 4 Number Negative Reciprocal 13 7 - 7 13 1 a - a

Example 5 The slope of y = -4x + 9 is -4. Find the slope of a line perpendicular to it: 1 4 m = The perpendicular line has a slope of ¼, and goes through the point (2, 7).

The lines are neither parallel nor perpendicular. Example 6 2x + 5y = 9 4x + y = 1 y = -4x + 1 m2 = -4 y = - x + 2 5 9 2 5 m1 = - The lines are neither parallel nor perpendicular.

Homework: pp. 259-261