LINEAR EQUATIONS

MSJC ~ Menifee Valley CampusMath Center Workshop SeriesJanice Levasseur

Equations of the LineWrite 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 pointWrite 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 constantsGiven the y-int = (0, -3/2) b = - 3/2Given the slope = 6 m = 6Putting everything together we get the equation of the line in slope-int form:y=mx+by = 6x 3/2

Ex: Find an equation of the line with slope = 1.23 and y-int = (0, 0.63)Recall: slope-intercept form of a linear equation y = mx + b, where m and b are constantsGiven the y-int = (0, 0.63) b = 0.63Given the slope = 1.23 m = 1.23Putting everything together we get the equation of the line in slope-int form:y=mx+by = 1.23x + 0.63

Equations of the LineWrite the equation of a line given the slope and the y-interceptWrite 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 containsthe point (4, 2) Start with the slope-intercept form of a linear equation y = mx + bSlope = - 3 y = - 3x + bWhat is b, though?What is b, though?Use the given point (4, 2) x = 4 and y = 2y = - 3x + b 2 = - 3(4) + b2 = -12 + b 14 = b put it together we have m and b y = - 3 x + 14

Ex: Find an equation of the line with slope = -0.25 that contains the point (2, -6) Start with the slope-intercept form of a linear equation y = mx + bSlope = -0.25 y = -0.25 x + bWhat is b, though?Use the given point (2, -6) x = 2 and y = -6y = -0.25 x + b -6 = -0.25(2) + b-6 = -0.5 + b -5.5 = b put it together we have m and b y = -0.25x 5.5

Equations of the LineWrite the equation of a line given the slope and the y-interceptWrite the equation of a line given the slope and a pointWrite 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)First, find the slope of the line containing the points:Slope = m = rise = y1 - y2 = 1 (5) run x1 - x2 -2 3 Point 1Point 2Now we have m = 4/5 and two points. Pick one point and proceed like in the last section.

We have m = 4/5, the point (-2, 1), and y = mx + bSlope = 4/5 y = 4/5x + bWhat is b, though?Use the given point (-2, 1) x = -2 and y = 1y = 4/5x + b 1 = 4/5(-2) + b1 = (-8/5) + b 13/5 = b put it together we have m and b y = 4/5x + 13/5

Ex: Find an equation of the line containing the points (-4, 5) and (-2, -3)First, find the slope of the line containing the points:Slope = m = rise = y1 - y2 = 5 (-3) run x1 - x2 -4 (-2) Point 1Point 2Now we have m = -4 and two points. Pick one point and proceed like in the last section.= -4

We have m = -4, the point (-4, 5), and y = mx + bSlope = -4 y = -4x + bWhat is b, though?Use the given point (-4, 5) x = -4 and y = 5y = -4x + b 5 = -4(-4) + b5 = 16 + b -11 = b put it together we have m and b y = -4x 11

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) Point 1Point 2Now we have m = -5 and two points. Pick one point and proceed like in the last section.= -5

We have m = -5, the point (0, 0), and y = mx + bSlope = -5 y = -5x + bWhat is b, though?Use the given point (0, 0) x = 0 and y = 0y = -5x + b 0 = -5(0) + b0 = 0 + b 0 = b put it together we have m and b y = -5x + 0 y = -5x

Equations of the LineWrite the equation of a line given the slope and the y-interceptWrite the equation of a line given the slope and a pointWrite the equation of a line given two points

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.

Perpendicular Lines (Situation 1)Perpendicular lines intersect at a right angle Notation: L1: y = m1x + b1L2: y = m2x + b2L1 ^ L2

Nonvertical perpendicular lines have slopes that are the negative reciprocals of each other: m1m2 = -1 ~ or ~ m1 = - 1/m2 ~ or ~ m2 = - 1/m1If 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 = - 1First, determine the slopes of each line by rewriting the equations in slope-intercept form:l1: y = - x + 8 and l2: y = x + 1m1 = and m2 = 11-11Since 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 = 16First, 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 + 8m1 = and m2 = 2/33/2Since m1m2 = (2/3)(3/2) = 1 = -1Therefore, the lines are not perpendicular!

Parallel Lines (Situation 2)Parallel lines do not intersect Notation: L1: y = m1x + b1L2: y = m2x + b2L1 || L2

Nonvertical parallel lines have the same slopes but different y-intercepts: m1 = m2 ~ and ~ b1 = b2Horizontal 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 = - 2First, determine the slopes and intercepts of each line by rewriting the equations in slope-intercept form:l1: y = 3x + 5 and l2: y = 3x - 2m1 = and m2 = 33Since m1 = m2 and b1 = b2the lines are parallel.b1 = and b2 = 5-2

Ex: Determine whether or not the graphs of the equations of the lines are parallel:l1: 4x + y = 3 and l2: x + 4y = - 4First, determine the slopes and intercepts of each line by rewriting the equations in slope-intercept form:l1: y = -4x + 3 and l2: y = (-)x - 1m1 = and m2 = -4- Since m1 = m2the lines are not parallel.

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