Slope-Intercept Form of the Equation of a Line The slope-intercept equation of a non-vertical line with slope m and y-intercept b is y = mx + b. Point-Slope Form of the Equation of a Line The point-slope equation of a non-vertical line of slope m that passes through the point (x1, y1) is y – y1 = m(x – x1).

Example: Writing the Point-Slope Equation of a Line Write the point-slope form of the equation of the line passing through (-1,3) with a slope of 4. Then solve the equation for y. Solution We use the point-slope equation of a line with m = 4, x1= -1, and y1 = 3. This is the point-slope form of the equation. y – y1 = m(x – x1) Substitute the given values. Simply. y – 3 = 4[x – (-1)] We now have the point-slope form of the equation for the given line. y – 3 = 4(x + 1) We can solve the equation for y by applying the distributive property. y – 3 = 4x + 4 y = 4x + 7 Add 3 to both sides.

Steps for Graphing y = mx + b Graphing y = mx + b by Using the Slope and y-Intercept Plot the y-intercept on the y-axis. This is the point (0, b). Obtain a second point using the slope, m. Write m as a fraction, and use rise over run starting at the y-intercept to plot this point. Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.

Equations of Horizontal and Vertical Lines Equation of a Horizontal Line A horizontal line is given by an equation of the form y = b where b is the y-intercept. Note: m = 0. Equation of a Vertical Line A vertical line is given by an equation of the form x = a where a is the x-intercept. Note: m is undefined.

2-4 WRITING LINEAR EQUATIONS

Equations of the Line Write the equation of a line given the slope and the y-intercept Write the equation of a line given the slope and a point Write the equation of a line given two points

Equations of the Line Write the equation of a line given the slope and the y-intercept: m and (0, b) Write the equation of a line given the slope and a point Write the equation of a line given two points

Ex: Find an equation of the line with slope = 6 and y-int = (0, -3/2) Recall: slope-intercept form of a linear equation y = mx + b, where m and b are constants Given the y-int = (0, -3/2)  b = - 3/2 Given the slope = 6  m = 6

Equations of the Line Write the equation of a line given the slope and the y-intercept Write the equation of a line given the slope and a point: m and (x1, y1) Write the equation of a line given two points

Ex: Find an equation of the line with slope = -3 that contains the point (4, 2)    

Equations of the Line Write the equation of a line given the slope and the y-intercept Write the equation of a line given the slope and a point Write the equation of a line given two points: (x1, y1) and (x2, y2)

Ex: Find an equation of the line containing the points (-2, 1) and (3, 5)    

Ex: Find an equation of the line containing the points (-4, 5) and (-2, -3)    

Ex: Find an equation of the line containing the points (0, 0) and (1, -5) First, find the slope of the line containing the points: Slope = m = rise = y1 - y2 = 0 – (-5) run x1 - x2 0 – (1) = -5 Now we have m = -5 and two points. Pick one point and proceed like in the last section.

Equations of the Line Write the equation of a line given the slope and the y-intercept Write the equation of a line given the slope and a point Write the equation of a line given two points

We have m = -5, the point (0, 0), and y = mx + b Slope = -5  y = -5x + b What is b, though? Use the given point (0, 0)  x = 0 and y = 0 y = -5x + b  0 = -5(0) + b 0 = 0 + b 0 = b put it together  we have m and b  y = -5x + 0 y = -5x

Parallel & Perpendicular Lines When we graph a pair of linear equations, there are three possibilities: the graphs intersect at exactly one point the graphs do not intersect the graphs intersect at infinitely many points We will consider a special case of situation 1 and also situation 2.

Slope and Parallel Lines If two non-vertical lines are parallel, then they have the same slope. If two distinct non-vertical lines have the same slope, then they are parallel. Two distinct vertical lines, both with undefined slopes, are parallel.

Slope and Perpendicular Lines 90° Two lines that intersect at a right angle (90°) are said to be perpendicular. There is a relationship between the slopes of perpendicular lines. Slope and Perpendicular Lines If two non-vertical lines are perpendicular, then the product of their slopes is –1. If the product of the slopes of two lines is –1, then the lines are perpendicular. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.

Perpendicular Lines (Situation 1) Perpendicular lines intersect at a right angle Notation: L1: y = m1x + b1 L2: y = m2x + b2 L1 ^ L2

Nonvertical perpendicular lines have slopes that are the negative reciprocals of each other: m1m2 = -1 ~ or ~ m1 = - 1/m2 ~ or ~ m2 = - 1/m1 If l1 is vertical (l1: x = a) and is perpendicular to l2, then l2 is horizontal (l2: y = b) ~ and ~ vice versa

Ex: Determine whether or not the graphs of the equations of the lines are perpendicular: l1: x + y = 8 and l2: x – y = - 1 First, determine the slopes of each line by rewriting the equations in slope-intercept form: l1: y = - x + 8 and l2: y = x + 1 1 1 m1 = and m2 = -1 1 Since m1m2 = (-1)(1) = -1, the lines are perpendicular.

Ex: Determine whether or not the graphs of the equations of the lines are perpendicular: l1: -2x + 3y = -21 and l2: 2y – 3x = 16 First, determine the slopes of each line by rewriting the equations in slope-intercept form: l1: y = (2/3)x - 7 and l2: y = (3/2)x + 8 m1 = and m2 = 2/3 3/2 Since m1m2 = (2/3)(3/2) = 1 = -1 Therefore, the lines are not perpendicular!

Parallel Lines (Situation 2) Parallel lines do not intersect Notation: L1: y = m1x + b1 L2: y = m2x + b2 L1 || L2

Nonvertical parallel lines have the same slopes but different y-intercepts: m1 = m2 ~ and ~ b1 = b2 Horizontal Parallel Lines have equations y = p and y = q where p and q differ. Vertical Parallel Lines have equations x = p and x = q where p and q differ.

Ex: Determine whether or not the graphs of the equations of the lines are parallel: l1: 3x - y = -5 and l2: y – 3x = - 2 First, determine the slopes and intercepts of each line by rewriting the equations in slope-intercept form: l1: y = 3x + 5 and l2: y = 3x - 2 m1 = and m2 = 3 3 b1 = and b2 = 5 -2 Since m1 = m2 and b1 = b2 the lines are parallel.

Ex: Determine whether or not the graphs of the equations of the lines are parallel: l1: 4x + y = 3 and l2: x + 4y = - 4 First, determine the slopes and intercepts of each line by rewriting the equations in slope-intercept form: l1: y = -4x + 3 and l2: y = (-¼)x - 1 m1 = and m2 = -4 - ¼ Since m1 = m2 the lines are not parallel.

Problems Homework pg , numbers Find the slope of the line that is a) parallel b) perpendicular to the given lines. y = 3x 8x + y = 11 3x – 4y + 7 = 0 y = 9 2. Write the equation for each line in slope-intercept form. Passes thru (-2,-7) and parallel to y = -5x+4 Passes thru (-4, 2) and perpendicular to y = x/3 + 7 Homework pg , numbers

HW: 2-4 WORKSHEET Front and Back (Due Next Time)