Algebra 2 by Richard Wright

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0-04 Find Slope and Write Equations of Lines

Mr. Wright teaches the lesson.

Objectives:

SDA NAD Content Standards (2018): AII.4.1, AII.4.2, AII.5.1, AII.5.3, AII.7.1

mountain
Figure 1: Mountain (Pixabay/BitBot-Media)

The steepness of a side of a mountainside is the slope. Skiers like to ski down steep slopes.

Slope

Slope describes how steep a graph of a line is from left to right. It is how much the y changes as the x changes.

slope
Figure 2: \(Slope = \frac{rise}{run}\)

$$ Slope = \frac{rise}{run} $$

$$ m = \frac{y_2-y_1}{x_2-x_1} $$

In real-life problems, slope is the same thing as rate of change because rate of change is how much one quantity changes as another changes. Think of a graph where the y-axis is measured in dollars and the x-axis is measured in years. The slope formula is y divided by x, so the new units would be dollars per year which is the rate by the which the amount of dollars changes every year.

As you move from left to right, a positive slope rises, a negative slope falls, zero slope is horizontal, and no slope (or undefined slope) is vertical.

types of slope
Figure 3: Types of slope
Slope of Lines

$$ Slope = \frac{rise}{run} $$

$$ m = \frac{y_2-y_1}{x_2-x_1} $$

Example 1

Find the slope of the line passing through the given points. Classify as rises, falls, horizontal, or vertical.

  1. (1, 4), (3, 8)
  2. (5, 2), (–3, 5)
  3. (3, 1), (–2, 1)

Solution

Fill in the slope formula with the points. Remember the y's go on top.

  1. (1, 4), (3, 8)

    $$ m=\frac{y_2-y_1}{x_2-x_1} $$

    $$ m=\frac{8-4}{3-1} $$

    $$ m=\frac{4}{2}=2 $$

    It is a positive slope, so it rises.

  2. (5, 2), (–3, 5)

    $$ m=\frac{y_2-y_1}{x_2-x_1} $$

    $$ m=\frac{5-2}{-3-5} $$

    $$ m=-\frac{3}{8} $$

    It is a negative slope, so it falls.

  3. (3, 1), (–2, 1)

    $$ m=\frac{y_2-y_1}{x_2-x_1} $$

    $$ m=\frac{1-1}{-2-3} $$

    $$ m=\frac{0}{-5}=0 $$

    It is zero slope, so it is horizontal.

Example 2

scuba diver
Figure 4: Scuba diver (pixabay/joakant)

A scuba diver is 30 feet below the surface of the water 1 minute after he entered the water and 100 feet below the surface after 2.5 minutes. The suggested rate of change of depth is 30 ft/min. Is this diver following the recommendation?

Solution

Rate of change is the same as slope. Time is usually the independent, or x, variable to the depth will be the y variable. The two points would be (1, 30) and (2.5, 100).

$$ m=\frac{y_2-y_1}{x_2-x_1} $$

$$ m=\frac{100-30}{2.5-1}=\frac{70}{1.5}\approx46.7 $$

Since this is a word problem, the answer wanted is probably a decimal, so the diver’s rate of change of depth is 46.7 ft/min. Thus the diver is not following the recommendation.

Parallel and Perpendicular Lines

Parallel lines are lines in the same plane that do not intersect. They go the same direction. Parallel lines have the same slope.

parallel lines
Figure 5: Parallel Lines

Perpendicular lines are in the same plane that intersect to form a right angle. The product of the slopes of perpendicular lines is −1. This means the slopes are negative reciprocals such as \(\frac{2}{3}\) and \(-\frac{3}{2}\).

perpendicular lines
Figure 6: Perpendicular Lines
Parallel and Perpendicular Lines

Parallel line have the same slope.

m1 = m2

Perpendicular lines have slope whose product is –1. The slope are also negative reciprocals.

m1 · m2 = –1 or \(m_2 = -\frac{1}{m_1}\)

Example 3

Tell whether the lines are parallel, perpendicular, or neither.

  1. Line 1: through (–2, 8) and (2, –4)
    Line 2: through (–5, 1) and (–2, 2)
  2. Line 1: through (–4, –2) and (1, 7)
    Line 2: through (–1, –4) and (3, 5)

Solution

Find the slopes and compare them.

  1. $$ \begin{array}{lrl}\text{Line 1:} & m_1=\frac{-4-8}{2-\left(-2\right)} & =-\frac{12}{4} = -3 \\ \text{Line 2:} & m_2=\frac{2-1}{-2-\left(-5\right)} & =\frac{1}{3} \end{array} $$

    \(-3\cdot\frac{1}{3}=-1\) so the lines are perpendicular.

  2. $$ \begin{array}{lrl} \text{Line 1:} & m_1=\frac{7-\left(-2\right)}{1-\left(-4\right)} & =\frac{9}{5} \\ \text{Line 2:} & m_2=\frac{5-\left(-4\right)}{3-\left(-1\right)} & =\frac{9}{4} \end{array} $$

    Since the slopes are not the same, nor is their product −1, the lines are neither.

Write Equations of Lines

Write Equations of Lines

To write equations of lines:

  1. Find the slope from the two points.
  2. Fill the slope, m, and one point, (x, y) into y = mx + b and solve for b.
  3. Fill m and b into y = mx + b.

Example 4

Find the equation of the line through (1, 2) and (–2, 4).

Solution

Find the slope:

$$ m=\frac{y_2-y_1}{x_2-x_1} $$

$$ m=\frac{4-2}{-2-1}=-\frac{2}{3} $$

Fill the slope, \(-\frac{2}{3}\), and a point (1, 2) into y = mx + b and solve forb.

$$ 2=-\frac{2}{3}\left(1\right)+b $$

$$ 2=-\frac{2}{3}+b $$

$$ 2+\frac{2}{3}=b $$

$$ b=\frac{6}{3}+\frac{2}{3}=\frac{8}{3} $$

Fill m and b into y = mx +b.

$$ y=-\frac{2}{3}x+\frac{8}{3} $$

Example 5

Write the equation of the line given in the graph.

line
Figure 7

Solution

The slope can be found from the graph. Counting rise and run from (2, −2) to (3, 1) gives a rise of 3 and a run of 1.

$$ slope = m = \frac{rise}{run} = \frac{3}{1} = 3 $$

Fill the slope, 3, and a point (3, 1) into y = mx + b and solve forb.

1 = 3(3) + b

−8 = b

Fill m and b into y = mx +b.

y = 3x − 8

Practice Exercises

    Find the slope of the line passing through the given points. Classify as rises, falls, horizontal, or vertical.

  1. (2, 6), (1, 7)
  2. (−1, 0), (2, 5)
  3. (7, 1), (7, 4)
  4. (3, 2), (−5, 2)

    5. Tell whether the lines are parallel, perpendicular, or neither.

  5. Line 1: through (1, 2) and (3, 4)
    Line 2: through (−2, 4) and (−3, 3)
  6. Line 1: through (0, 2) and (−5, 7)
    Line 2: through (−2, 3) and (0, 5)
  7. Line 1: through (5, 5) and (1, 0)
    Line 2: through (1, 10) and (9, 0)

    8. Write the equation of the line with the given information.

  8. Slope = 3, passes through (0, 3)
  9. Slope = −1, passes through (2, −1)
  10. Passes through (3, 2) and (1, −2)
  11. Passes through (2, 0) and (0, 4)
  12. Has the graph
    line

    13. Problem Solving

  13. Two points on a line are (1, 3) and (4, k). Find the value of k so that the slope is 2.
  14. A hiker is climbing up a steep hill to an overlook. After 10 minutes he is at an altitude of 700 ft. Then at 30 minutes he is at 850 ft. What is the hiker's rate of change of altitude?
  15. Jonah
    Figure 8: Jonah preaching at Nineveh (Otto Semler)
    In the book of Jonah, he preached in Nineveh for 40 days. a) Read Jonah 1:1, why did God send Jonah to Nineveh? b) Read Jonah 3:3, how long did it take to walk through the city? (The population was estimated at 120,000.) c) Read Jonah 3:4, how long did Nineveh have before God would overthrow them? d) If Jonah preached for 40 days and converted 120,000 people, what is the rate of change of people?

    16. Mixed Review

  16. (0-03) Cereal is sold by weight with a tolerance of 5% of the standard weight. Write and solve an absolute value inequality that describes the actual weights of a box of cereal labeled 14 oz.
  17. (0-03) Solve: |3x + 7| = 15
  18. (0-02) Joe has 50 feet of fence to go around his garden to keep rabbits out. The garden is 8 feet long. a) Which problem solving strategy will you use to find the width? b) Write an equation to solve for the width. c) Solve the equation.
  19. (0-01) Solve for ℓ: SA = πr2 + πrℓ
  20. (0-01) Solve: 3(x + 1) ≤ 7(x − 3)

Answers

  1. −1, falls
  2. \(\frac{5}{3}\), rises
  3. No slope, vertical
  4. 0, horizontal
  5. Parallel
  6. Perpendicular
  7. Neither
  8. y = 3x + 3
  9. y = −x + 1
  10. y = 2x − 4
  11. y = −2x + 4
  12. y = −4x − 8
  13. 9
  14. 7.5 ft/min
  15. d) 3000 people/day
  16. |x − 14| ≤ 0.05(14); 13.3 ≤ x ≤ 14.7
  17. \(-\frac{22}{3}\), \(\frac{8}{3}\)
  18. Formula, P = 2ℓ + 2w, 17 ft
  19. \(ℓ=\frac{SA-πr^2}{πr}\)
  20. x ≥ 6