Section 1.10: Lines We specifically study lines and their equations in this section.

Slope of a Line The slope m of the line passing through the points (x1 , y1 ) and (x2 , y2 ), with x1 ̸= x2 , is given by y2 − y1 m= . x 2 − x1

Exercises: Find the slope of the line passing through the given points, if possible. i) (1, 2) and (5, 3)

ii) (−6, 2) and (−4, 3)

Slope-Intercept Form of the Equation of a Line An equation of a line that has slope m and y-intercept b is y = mx + b.

Equations of lines are not at all difficult to graph by hand when in slope-intercept form. Consider the linear equation y = 2x − 1. To graph it, we do the following.

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We first identify the y-intercept, which is (0, −1), and we plot it.

Then we identify slope, which in this case is 2 or 21 . Using the y-intercept as our starting position, we move up 2 units then over 1 unit and plot a second point at (1, 1).

Finally, we draw a straight line to connect the points.

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Exercises: Find the slope and y-intercept for each equation (if possible). Then sketch the graph that represents the equations. i) 5x + 25y = 50

ii) 6x − 3y = 12

iii) y = 6

iv) x = −2

Point-Slope Form of the Equation of a Line An equation of the line with slope m passing through the point (x1 , y1 ) is y − y1 = m(x − x1 ).

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If we are given a point and a slope, we can find the equation of the line passing through the given point and having that slope with this formula.

Exercises: Find the equation of the line passing through the given point and having the given slope. If possible, write in slope-intercept form. i) (1, 3);

m=5

Solution. We know the slope: m = 5. And we let x1 = 1 and y1 = 3. Using the point-slope form we have y − 3 = 5(x − 1) . We then rewrite this expression in slope-intercept form. So, we have y − 3 = 5x − 5 =⇒ y = 5x − 2 . ii) (2, 4);

iii) (1/3, 21);

m=0

m undefined

If we are given two points, we can find the equation of the line passing through both given points. Ultimately, we will use the point-slope form. We already have a point to work with. All we need is a slope. Can we find the slope of the line passing through two points? Exercise: Find the slope-intercept form of the equation of the line passing through the points (1, 4) and (2, 1).

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Characterization of Parallel and Perpendicular Lines Two distinct lines with slopes m1 and m2 , neither of which is vertical, are parallel if and only if m1 = m2 . Two distinct lines with nonzero slopes m1 and m2 are perpendicular if and only if 1 m1 m2 = −1, that is, m1 = − m . 2 Given the equation of a line L1 , we may find a line parallel or perpendicular to L1 that passes through the point (x1 , y1 ) using the point-slope form. Remember: All we need is a point and a slope. Examples: i) Find the equation of the line parallel to the line representing y = − 12 x + 4 and passing through (1, 2).

ii) Find the equation of the line perpendicular to the line representing y = 13 x + 5 passing through (−1, 3).

Additional Practice pp. 120-121 Nos. 1-51, 57, 62, 68, 71

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