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LINEAR EQUATIONS MSJC ~ Menifee Valley Campus Math Center Workshop Series Janice Levasseur

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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

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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

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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 Putting everything together we get the equation of the line in slope-int form: y = mx+b6- 3/2 y = 6x – 3/2

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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 constants Given the y-int = (0, 0.63) b = 0.63 Given the slope = 1.23 m = 1.23 Putting everything together we get the equation of the line in slope-int form: y = m x +b 1.230.63 y = 1.23x + 0.63

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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 (x 1, y 1 ) Write the equation of a line given two points

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Ex: Find an equation of the line with slope = -3 that contains the point (4, 2) Start with the slope-intercept form of a linear equation y = mx + b Slope = - 3 y = - 3x + bWhat is b, though? Use the given point (4, 2) x = 4 and y = 2 y = - 3x + b 2 = - 3(4) + b 2 = -12 + b 14 = b put it together we have m and b y = - 3 x + 14

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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 + b Slope = -0.25 y = -0.25 x + bWhat is b, though? Use the given point (2, -6) x = 2 and y = -6 y = -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

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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: (x 1, y 1 ) and (x 2, y 2 )

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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 = y 1 - y 2 = 1 – (5) run x 1 - x 2 -2 – 3 Point 1Point 2 Now we have m = 4/5 and two points. Pick one point and proceed like in the last section.

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

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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 = y 1 - y 2 = 5 – (-3) run x 1 - x 2 -4 – (-2) Point 1Point 2 Now we have m = -4 and two points. Pick one point and proceed like in the last section. = -4

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

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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 = y 1 - y 2 = 0 – (-5) run x 1 - x 2 0 – (1) Point 1Point 2 Now we have m = -5 and two points. Pick one point and proceed like in the last section. = -5

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We have m = -5, the point (0, 0), and y = mx + b Slope = -5 y = -5x + bWhat 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

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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

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Parallel & Perpendicular Lines When we graph a pair of linear equations, there are three possibilities: 1. the graphs intersect at exactly one point 2. the graphs do not intersect 3. the graphs intersect at infinitely many points We will consider a special case of situation 1 and also situation 2.

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Perpendicular Lines (Situation 1) Perpendicular lines intersect at a right angle Notation: ›L 1 : y = m 1 x + b 1 ›L 2 : y = m 2 x + b 2 ›L 1 L 2

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Nonvertical perpendicular lines have slopes that are the negative reciprocals of each other: m 1 m 2 = -1 ~ or ~ m 1 = - 1/m 2 ~ or ~ m 2 = - 1/m 1 If l 1 is vertical (l 1 : x = a) and is perpendicular to l2, then l 2 is horizontal (l 2 : y = b) ~ and ~ vice versa

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Ex: Determine whether or not the graphs of the equations of the lines are perpendicular: l 1 : x + y = 8 and l 2 : x – y = - 1 First, determine the slopes of each line by rewriting the equations in slope-intercept form: l 1 : y = - x + 8 and l 2 : y = x + 1 m 1 = and m 2 = 11 1 Since m 1 m 2 = (-1)(1) = -1, the lines are perpendicular.

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Ex: Determine whether or not the graphs of the equations of the lines are perpendicular: l 1 : -2x + 3y = -21 and l 2 : 2y – 3x = 16 First, determine the slopes of each line by rewriting the equations in slope-intercept form: l 1 : y = (2/3)x - 7 and l 2 : y = (3/2)x + 8 m 1 = and m 2 =2/33/2 Since m 1 m 2 = (2/3)(3/2) = 1 = -1 Therefore, the lines are not perpendicular!

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Parallel Lines (Situation 2) Parallel lines do not intersect Notation: ›L 1 : y = m 1 x + b 1 ›L 2 : y = m 2 x + b 2 ›L 1 L 2

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Nonvertical parallel lines have the same slopes but different y-intercepts: m 1 = m 2 ~ and ~ b 1 = b 2 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.

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Ex: Determine whether or not the graphs of the equations of the lines are parallel: l 1 : 3x - y = -5 and l 2 : y – 3x = - 2 First, determine the slopes and intercepts of each line by rewriting the equations in slope-intercept form: l 1 : y = 3x + 5 and l 2 : y = 3x - 2 m 1 = and m 2 =33 Since m 1 = m 2 and b 1 = b 2 the lines are parallel. b 1 = and b 2 =5-2

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Ex: Determine whether or not the graphs of the equations of the lines are parallel: l 1 : 4x + y = 3 and l 2 : x + 4y = - 4 First, determine the slopes and intercepts of each line by rewriting the equations in slope-intercept form: l 1 : y = -4x + 3 and l 2 : y = (-¼)x - 1 m 1 = and m 2 =-4- ¼ Since m 1 = m 2 the lines are not parallel.

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