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Lines and slopes practice problems

Slopes, and equations of lines through points, parallel or perpendicular to a given line. 63 problems with worked solutions; in 63 of them every equation is proved by a computer algebra system.

16 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 13.124

easy
Find the equation of the line through \( \displaystyle (-4, 4) \) that is perpendicular to \( \displaystyle y = 1 - 4 x \), in slope-intercept form.
  1. \[ - \frac{1}{-4} = \frac{1}{4} \]
    Perpendicular slopes are negative reciprocals.✓ Proved
  2. \[ \frac{x + 4}{4} + 4 = \frac{x}{4} + 5 \]
    Point-slope form, then simplify.✓ Proved
  3. \[ \left. \frac{x}{4} + 5 \right|_{\substack{ x=-4 }} = 4 \]
    The line passes through (-4, 4).✓ Proved
Answer \( y = \frac{x}{4} + 5 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.131

easy
Find the slope of the line through \( \displaystyle (-5, -1) \) and \( \displaystyle (-4, 4) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
  1. \[ \frac{1 + 4}{-4 + 5} = 5 \]
    Rise over run.✓ Proved
  2. \[ 5 \left(x + 5\right) - 1 = 5 x + 24 \]
    Point-slope form through (-5, -1), then simplify.✓ Proved
  3. \[ \left. 5 x + 24 \right|_{\substack{ x=-4 }} = 4 \]
    It passes through (-4, 4) too.✓ Proved
Answer \( m = 5\ (\text{increasing}),\quad y = 5 x + 24 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.134

easy
Find the equation of the line through \( \displaystyle (5, -2) \) that is perpendicular to \( \displaystyle y = - \frac{5 x}{2} - 4 \), in slope-intercept form.
  1. \[ - \frac{1}{\left(-1\right) 5 \cdot \frac{1}{2}} = \frac{2}{5} \]
    Perpendicular slopes are negative reciprocals.✓ Proved
  2. \[ \frac{2 \left(x - 5\right)}{5} - 2 = \frac{2 x}{5} - 4 \]
    Point-slope form, then simplify.✓ Proved
  3. \[ \left. \frac{2 x}{5} - 4 \right|_{\substack{ x=5 }} = -2 \]
    The line passes through (5, -2).✓ Proved
Answer \( y = \frac{2 x}{5} - 4 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.296

easy
Find the slope of the line through \( \displaystyle (-5, -1) \) and \( \displaystyle (4, -2) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
  1. \[ \frac{-2 + 1}{4 + 5} = - \frac{1}{9} \]
    Rise over run.✓ Proved
  2. \[ \frac{- x - 5}{9} - 1 = - \frac{x}{9} - \frac{14}{9} \]
    Point-slope form through (-5, -1), then simplify.✓ Proved
  3. \[ \left. \frac{\left(-1\right) x}{9} - \frac{14}{9} \right|_{\substack{ x=4 }} = -2 \]
    It passes through (4, -2) too.✓ Proved
Answer \( m = - \frac{1}{9}\ (\text{decreasing}),\quad y = - \frac{x}{9} - \frac{14}{9} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.297

easy
Find the equation of the line through \( \displaystyle (-5, 5) \) that is parallel to \( \displaystyle y = \frac{2 x}{3} + 3 \), in slope-intercept form.
  1. Parallel lines share a slope. The slope is 2/3.
    Reviewed
  2. \[ \frac{2 \left(x + 5\right)}{3} + 5 = \frac{2 x}{3} + \frac{25}{3} \]
    Point-slope form, then simplify.✓ Proved
  3. \[ \left. \frac{2 x}{3} + \frac{25}{3} \right|_{\substack{ x=-5 }} = 5 \]
    The line passes through (-5, 5).✓ Proved
Answer \( y = \frac{2 x}{3} + \frac{25}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.342

easy
Find the slope of the line through \( \displaystyle (-5, -5) \) and \( \displaystyle (-2, -3) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
  1. \[ \frac{-3 + 5}{-2 + 5} = \frac{2}{3} \]
    Rise over run.✓ Proved
  2. \[ \frac{2 \left(x + 5\right)}{3} - 5 = \frac{2 x}{3} - \frac{5}{3} \]
    Point-slope form through (-5, -5), then simplify.✓ Proved
  3. \[ \left. \frac{2 x}{3} - \frac{5}{3} \right|_{\substack{ x=-2 }} = -3 \]
    It passes through (-2, -3) too.✓ Proved
Answer \( m = \frac{2}{3}\ (\text{increasing}),\quad y = \frac{2 x}{3} - \frac{5}{3} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.344

easy
Find the equation of the line through \( \displaystyle (5, -5) \) that is perpendicular to \( \displaystyle y = 1 - \frac{4 x}{3} \), in slope-intercept form.
  1. \[ - \frac{1}{\left(-1\right) 4 \cdot \frac{1}{3}} = \frac{3}{4} \]
    Perpendicular slopes are negative reciprocals.✓ Proved
  2. \[ \frac{3 \left(x - 5\right)}{4} - 5 = \frac{3 x}{4} - \frac{35}{4} \]
    Point-slope form, then simplify.✓ Proved
  3. \[ \left. \frac{3 x}{4} - \frac{35}{4} \right|_{\substack{ x=5 }} = -5 \]
    The line passes through (5, -5).✓ Proved
Answer \( y = \frac{3 x}{4} - \frac{35}{4} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.348

easy
Find the equation of the line through \( \displaystyle (-3, -3) \) that is perpendicular to \( \displaystyle y = 4 x + 1 \), in slope-intercept form.
  1. Perpendicular slopes are negative reciprocals: -1/4.
    Reviewed
  2. \[ \frac{- x - 3}{4} - 3 = - \frac{x}{4} - \frac{15}{4} \]
    Point-slope form, then simplify.✓ Proved
  3. \[ \left. \frac{\left(-1\right) x}{4} - \frac{15}{4} \right|_{\substack{ x=-3 }} = -3 \]
    The line passes through (-3, -3).✓ Proved
Answer \( y = - \frac{x}{4} - \frac{15}{4} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.421

easy
Find the slope of the line through \( \displaystyle (1, 3) \) and \( \displaystyle (-1, 1) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
  1. \[ \frac{-3 + 1}{-1 - 1} = 1 \]
    Rise over run.✓ Proved
  2. \[ 1 \left(x - 1\right) + 3 = x + 2 \]
    Point-slope form through (1, 3), then simplify.✓ Proved
  3. \[ \left. x + 2 \right|_{\substack{ x=-1 }} = 1 \]
    It passes through (-1, 1) too.✓ Proved
Answer \( m = 1\ (\text{increasing}),\quad y = x + 2 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.423

easy
Find the equation of the line through \( \displaystyle (2, -1) \) that is parallel to \( \displaystyle y = x - 1 \), in slope-intercept form.
  1. Parallel lines share a slope. The slope is 1.
    Reviewed
  2. \[ 1 \left(x - 2\right) - 1 = x - 3 \]
    Point-slope form, then simplify.✓ Proved
  3. \[ \left. x - 3 \right|_{\substack{ x=2 }} = -1 \]
    The line passes through (2, -1).✓ Proved
Answer \( y = x - 3 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.426

easy
Find the slope of the line through \( \displaystyle (-5, 3) \) and \( \displaystyle (1, -2) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
  1. \[ \frac{-3 - 2}{1 + 5} = - \frac{5}{6} \]
    Rise over run.✓ Proved
  2. \[ 3 - \frac{5 \left(x + 5\right)}{6} = - \frac{5 x}{6} - \frac{7}{6} \]
    Point-slope form through (-5, 3), then simplify.✓ Proved
  3. \[ \left. - \frac{5 x}{6} - \frac{7}{6} \right|_{\substack{ x=1 }} = -2 \]
    It passes through (1, -2) too.✓ Proved
Answer \( m = - \frac{5}{6}\ (\text{decreasing}),\quad y = - \frac{5 x}{6} - \frac{7}{6} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.499

easy
Find the equation of the line through \( \displaystyle (-2, 2) \) that is perpendicular to \( \displaystyle y = - 4 x - 4 \), in slope-intercept form.
  1. \[ - \frac{1}{-4} = \frac{1}{4} \]
    Perpendicular slopes are negative reciprocals.✓ Proved
  2. \[ \frac{x + 2}{4} + 2 = \frac{x}{4} + \frac{5}{2} \]
    Point-slope form, then simplify.✓ Proved
  3. \[ \left. \frac{x}{4} + \frac{5}{2} \right|_{\substack{ x=-2 }} = 2 \]
    The line passes through (-2, 2).✓ Proved
Answer \( y = \frac{x}{4} + \frac{5}{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.502

easy
Find the slope of the line through \( \displaystyle (-5, 2) \) and \( \displaystyle (-2, -4) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
  1. \[ \frac{-4 - 2}{-2 + 5} = -2 \]
    Rise over run.✓ Proved
  2. \[ 2 - 2 \left(x + 5\right) = - 2 x - 8 \]
    Point-slope form through (-5, 2), then simplify.✓ Proved
  3. \[ \left. - 2 x - 8 \right|_{\substack{ x=-2 }} = -4 \]
    It passes through (-2, -4) too.✓ Proved
Answer \( m = -2\ (\text{decreasing}),\quad y = - 2 x - 8 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.506

easy
Find the equation of the line through \( \displaystyle (-2, 5) \) that is parallel to \( \displaystyle y = 4 x - 3 \), in slope-intercept form.
  1. Parallel lines share a slope. The slope is 4.
    Reviewed
  2. \[ 4 \left(x + 2\right) + 5 = 4 x + 13 \]
    Point-slope form, then simplify.✓ Proved
  3. \[ \left. 4 x + 13 \right|_{\substack{ x=-2 }} = 5 \]
    The line passes through (-2, 5).✓ Proved
Answer \( y = 4 x + 13 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.121

easy
Find the equation of the line with slope \( \displaystyle \frac{5}{2} \) through \( \displaystyle (4, 2) \), in slope-intercept form.
  1. \[ \frac{5 \left(x - 4\right)}{2} + 2 = \frac{5 x}{2} - 8 \]
    Point-slope form, then simplify.✓ Proved
  2. \[ \left. \frac{5 x}{2} - 8 \right|_{\substack{ x=4 }} = 2 \]
    The line passes through (4, 2).✓ Proved
Answer \( y = \frac{5 x}{2} - 8 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.507

easy
Find the equation of the line with slope \( \displaystyle \frac{2}{3} \) through \( \displaystyle (-4, 4) \), in slope-intercept form.
  1. \[ \frac{2 \left(x + 4\right)}{3} + 4 = \frac{2 x}{3} + \frac{20}{3} \]
    Point-slope form, then simplify.✓ Proved
  2. \[ \left. \frac{2 x}{3} + \frac{20}{3} \right|_{\substack{ x=-4 }} = 4 \]
    The line passes through (-4, 4).✓ Proved
Answer \( y = \frac{2 x}{3} + \frac{20}{3} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More lines and slopes practice

Find the slope of the line through \( \displaystyle (3, -4) \) and \( \displaystyle (-3, 3) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.122easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (0, -2) \) that is perpendicular to \( \displaystyle y = - \frac{x}{3} - 3 \), in slope-intercept form.
Problem 13.123easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (-5, 5) \) and \( \displaystyle (3, -5) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.125easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (-3, 0) \) and \( \displaystyle (4, -3) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.126easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (-3, 5) \) that is parallel to \( \displaystyle y = 4 x \), in slope-intercept form.
Problem 13.127easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (2, 1) \) that is perpendicular to \( \displaystyle y = 5 x \), in slope-intercept form.
Problem 13.128easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (1, -2) \) that is parallel to \( \displaystyle y = x \), in slope-intercept form.
Problem 13.129easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle - \frac{3}{2} \) through \( \displaystyle (3, 1) \), in slope-intercept form.
Problem 13.130easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (3, -2) \) and \( \displaystyle (2, 0) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.132easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle 2 \) through \( \displaystyle (-2, -2) \), in slope-intercept form.
Problem 13.133easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (4, -2) \) that is parallel to \( \displaystyle y = 4 x - 1 \), in slope-intercept form.
Problem 13.135easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle 3 \) through \( \displaystyle (5, -4) \), in slope-intercept form.
Problem 13.136easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (-2, -5) \) that is parallel to \( \displaystyle y = \frac{5 x}{3} + 4 \), in slope-intercept form.
Problem 13.137easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle - \frac{2}{3} \) through \( \displaystyle (-5, 3) \), in slope-intercept form.
Problem 13.138easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (-3, -2) \) that is perpendicular to \( \displaystyle y = 1 - x \), in slope-intercept form.
Problem 13.139easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (0, 3) \) and \( \displaystyle (4, -2) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.140easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (3, -4) \) that is parallel to \( \displaystyle y = x - 4 \), in slope-intercept form.
Problem 13.141easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (-4, -5) \) that is parallel to \( \displaystyle y = \frac{5 x}{2} - 3 \), in slope-intercept form.
Problem 13.142easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (3, -5) \) and \( \displaystyle (0, 0) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.298easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (-2, 3) \) that is parallel to \( \displaystyle y = - \frac{x}{2} - 1 \), in slope-intercept form.
Problem 13.299easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (5, 5) \) and \( \displaystyle (-4, 0) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.300easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (2, -5) \) that is parallel to \( \displaystyle y = - \frac{2 x}{3} - 4 \), in slope-intercept form.
Problem 13.301easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (3, -1) \) that is parallel to \( \displaystyle y = 5 x \), in slope-intercept form.
Problem 13.302easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle \frac{5}{2} \) through \( \displaystyle (4, 5) \), in slope-intercept form.
Problem 13.303easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle - \frac{1}{3} \) through \( \displaystyle (1, -2) \), in slope-intercept form.
Problem 13.304easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (3, -4) \) that is perpendicular to \( \displaystyle y = - 2 x - 3 \), in slope-intercept form.
Problem 13.305easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (-3, 3) \) that is perpendicular to \( \displaystyle y = 3 x - 3 \), in slope-intercept form.
Problem 13.306easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle 4 \) through \( \displaystyle (-4, -5) \), in slope-intercept form.
Problem 13.307easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (2, -2) \) and \( \displaystyle (3, 2) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.308easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (-3, -3) \) that is parallel to \( \displaystyle y = - 4 x - 2 \), in slope-intercept form.
Problem 13.309easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle - \frac{3}{2} \) through \( \displaystyle (5, 2) \), in slope-intercept form.
Problem 13.310easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle 2 \) through \( \displaystyle (3, -3) \), in slope-intercept form.
Problem 13.343easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (-5, -4) \) that is parallel to \( \displaystyle y = \frac{3 x}{2} + 1 \), in slope-intercept form.
Problem 13.345easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (2, -2) \) that is parallel to \( \displaystyle y = x - 3 \), in slope-intercept form.
Problem 13.346easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle - \frac{5}{3} \) through \( \displaystyle (-2, 2) \), in slope-intercept form.
Problem 13.347easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (1, -1) \) and \( \displaystyle (4, -2) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.349easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (-1, 2) \) and \( \displaystyle (3, 1) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.350easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (-2, -2) \) that is parallel to \( \displaystyle y = - \frac{4 x}{3} - 2 \), in slope-intercept form.
Problem 13.422easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (3, -4) \) and \( \displaystyle (-1, 5) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.424easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (0, -1) \) that is parallel to \( \displaystyle y = \frac{3 x}{2} - 2 \), in slope-intercept form.
Problem 13.425easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (-2, -2) \) that is parallel to \( \displaystyle y = x - 3 \), in slope-intercept form.
Problem 13.427easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (-3, 4) \) and \( \displaystyle (-5, -1) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.428easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (5, 0) \) and \( \displaystyle (1, -4) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.500easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle - \frac{1}{2} \) through \( \displaystyle (4, -2) \), in slope-intercept form.
Problem 13.501easy✓ Every equation proved
Find the equation of the line with slope \( \displaystyle -4 \) through \( \displaystyle (0, 5) \), in slope-intercept form.
Problem 13.503easy✓ Every equation proved
Find the slope of the line through \( \displaystyle (-1, 5) \) and \( \displaystyle (-5, -3) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
Problem 13.504easy✓ Every equation proved
Find the equation of the line through \( \displaystyle (1, 4) \) that is perpendicular to \( \displaystyle y = 2 x - 3 \), in slope-intercept form.
Problem 13.505easy✓ Every equation proved